Stability of Inverse Problems for Steady Supersonic Flows Past Lipschitz Perturbed Cones

TL;DR

This paper investigates inverse problems in supersonic flows around cones, focusing on the stability of shocks and the existence of global solutions.

Contribution

The paper introduces a modified Glimm-type scheme and a functional to analyze the stability of oblique conical shocks in supersonic flows.

Findings

A modified Glimm-type scheme is developed to construct approximate solutions for inverse problems in supersonic flows.

The existence of global entropy solutions with bounded BV norm is established under certain flow conditions.

Entropy solutions asymptotically approach self-similar solutions determined by incoming flow and pressure.

Abstract

We are concerned with inverse problems for supersonic potential flows past infinite axisymmetric Lipschitz cones. The supersonic flows under consideration are governed by the steady isentropic Euler equations for axisymmetric potential flows, which give rise to a singular geometric source term. We first study the inverse problem for the stability of an oblique conical shock as an initial-boundary value problem with both the generating curve of the cone surface and the leading conical shock front as free boundaries. We then establish the existence and asymptotic behavior of global entropy solutions with bounded BV norm of the inverse problem, under the condition that the Mach number of the incoming flow is sufficiently large and the total variation of the pressure distribution on the cone is sufficiently small. To this end, we first develop a modified Glimm-type scheme to construct approximate solutions by self-similar solutions as building blocks to balance the influence of the geometric source term. Then we define a Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions. Meanwhile, the approximate generating curves of the cone surface are also constructed. Next, when the Mach number of the incoming flow is sufficiently large, by asymptotic analysis of the reflection coefficients in those interaction estimates, we prove that appropriate weights can be chosen so that the corresponding Glimm-type functional decreases in the flow direction. Finally, we determine the generating curves of the cone surface and establish the existence of global entropy solutions containing a strong leading conical shock, besides weak waves. Moreover, the entropy solution is proved to approach asymptotically the self-similar solution determined by the incoming flow and the asymptotic pressure on the cone surface at infinity.

Figures

The strong straight-sided conical shockFig. 2Supersonic flow past an axisymmetric cone

Interaction between weak waves

Reflection at the boundary

Near the strong shock wave

Full text

Introduction

We are interested in the structural stability of inverse problems for the three-dimensional (3-D) steady supersonic potential flows past a Lipschitz perturbed cone with given states of the incoming flow together with Lipschitz perturbed pressure distributions on its surface. The shock stability problem of steady supersonic flows past Lipschitz cones is fundamental for the mathematical theory of the multidimensional (M-D) hyperbolic systems of conservation laws, since its solutions are time-asymptotic states and global attractors of general entropy solutions of time-dependent initial-boundary value problems (IBVP) with abundant nonlinear phenomena, besides its significance to many fields of applications including aerodynamics; see [3, 9, 20, 29] and references cited therein. Meanwhile, the corresponding inverse problems play essential roles in airfoil design; see [1, 2, 5, 26–28, 39, 40, 43, 45]. As indicated in [20], when a uniform supersonic flow of constant speed from the far-field (negative infinity) hits a straight cone, given a constant pressure distribution that is less than a critical value on the cone surface, the vertex angle of the cone can be determined such that there is a supersonic straight-sided conical shock attached to the cone vertex, and the state between the conical shock-front and the cone can be obtained by the shooting method, which is a self-similar solution; see Fig. 1. In this paper, we focus our analysis on the stability of an inverse problem, along with the background self-similar solutions, in the steady potential flows that are axisymmetric with respect to the x–axis, given the pressure distributions of gas on the cones, whose boundary surfaces in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^3$$\end{document} , formed by the rotation of generating curves of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma := \{(x,b(x))\, : \, x\geqq 0\}$$\end{document} around the x–axis, are to be determined; see Fig. 2.Fig. 1. The strong straight-sided conical shockFig. 2Supersonic flow past an axisymmetric cone

To be more precise, the governing 3-D Euler equations for steady potential conical flows are of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&(\rho u)_x+(\rho v)_y=-\frac{\rho v}{y}, \\&v_x-u_y=0, \end{aligned} \right. \end{aligned}$$\end{document}

together with the Bernoulli law

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{u^2+v^2}{2}+\frac{c^2}{\gamma -1}= \frac{u_{\infty }^2}{2}+\frac{c_{\infty }^2}{\gamma -1}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U:=(u,v)^\top $$\end{document} is the velocity in the (x, y)–coordinates, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} is the flow density, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\infty }=(u_{\infty },0)^\top $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\infty }$$\end{document} are the velocity and the density of the incoming flow, respectively. The Bernoulli law in (1.2) is obtained via the constitutive relation between pressure p and density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} by scaling

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p=\rho ^{\gamma }, \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >1$$\end{document} for the polytropic isentropic gas and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1$$\end{document} for the isothermal flow. In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=:\sqrt{\frac{\gamma p}{\rho }}$$\end{document} is called the sonic speed, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:=\sqrt{\frac{u^2+v^2}{c^2}}$$\end{document} is called the Mach number.

The Bernoulli law (1.2) can be written as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{u^2+v^2}{2}+\frac{(u^2+v^2)M^{-2}}{\gamma -1}= \frac{u_\infty ^2}{2}+\frac{u_\infty ^2M_{\infty }^{-2}}{\gamma -1}. \end{aligned}$$\end{document}

Without loss of generality, we may choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\infty =1$$\end{document} by scaling; otherwise, we can simply scale: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U\rightarrow u_\infty ^{-1}U$$\end{document} , in system (1.1) and (1.3). With fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\infty =1$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }\rightarrow \infty $$\end{document} is equivalent to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\infty }\rightarrow 0$$\end{document} , or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{\infty }\rightarrow 0$$\end{document} .

System (1.1) can be written in the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _xW(U)+\partial _yH(U)=E(U,y) \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=(u,v)^\top $$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} W(U)=(\rho u,v)^\top ,\quad \,\, H(U)=(\rho v,-u)^\top ,\quad \,\, E(U,y)=(-\frac{\rho v}{y},0)^\top , \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} is a function of U determined by the Bernoulli law (1.2).

When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u>c$$\end{document} , U can also be presented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W(U)=(\rho u,v)^\top $$\end{document} , i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=U(W)$$\end{document} , by the implicit function theorem, since the Jacobian:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \det \big (\nabla _UW(U)\big )=-\frac{\rho }{c^2}(u^2-c^2)<0. \end{aligned}$$\end{document}

Regarding x as the time variable, (1.4) can be written as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _xW+\partial _yH(U(W))=E(U(W),y). \end{aligned}$$\end{document}

Therefore, system (1.1)–(1.2) becomes a hyperbolic system of conservation laws with source terms of form (1.5). Such nonhomogeneous hyperbolic systems of conservation laws also arise naturally in other problems from many important applications, which exhibit rich phenomena; for example, see [9, 11–13, 20, 23] and the references cited therein.

Throughout this paper, the following conditions are assumed:

• (A1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\,p^b(x)>0$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p^{b}(x)=p_{0}\qquad \text{ for } x\in [0,x_{0}], \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{0}>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0\in (0, p^*)$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^*>0$$\end{document} to be determined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma >1$$\end{document} , and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p^{b}\in \text {BV}([0,\infty )). \end{aligned}$$\end{document}

• (A2)   The velocity of the incoming flow \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\infty }=(1,0)^\top $$\end{document} is supersonic: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }>1$$\end{document} . Given a perturbed pressure distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^b(x)$$\end{document} on the cone surface, the problem is axisymmetric with respect to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x-$$\end{document} axis. Thus, it suffices to analyze the problem in the half-space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{y\leqq 0\}$$\end{document} . Then the inverse problem is to find the generating curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=b(x)\leqq 0$$\end{document} of the cone surface and a global solution in the domain:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega =\big \{(x,y)\,:\,x\geqq 0,y<b(x)\big \} \end{aligned}$$\end{document}

with its upper boundary:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma =\big \{(x,y)\,:\,x\geqq 0,y= b(x)\big \} \end{aligned}$$\end{document}

such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U\cdot {{\textbf {n}}}|_{\Gamma }=0, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {n}}}={{\textbf {n}}}\big (x,b(x)\big ) =\frac{(-b'(x),1)^\top }{\sqrt{1+(b'(x))^2}} $$\end{document} is the corresponding outer normal vector to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} at a differentiable point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x,b(x))\in \Gamma $$\end{document} .

With this setup, the inverse stability problem can be formulated into the following initial-boundary value problem (IBVP) for system (1.4):

Cauchy Condition:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U|_{x=0}=U_{\infty }:= (1,0)^\top , \end{aligned}$$\end{document}

Boundary Condition:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p\big (x,b(x)\big )=p^b(x). \end{aligned}$$\end{document}

We first introduce the notion of entropy solutions for problem (1.5)–(1.10).

Definition 1.1

(Entropy Solutions).   Consider the inverse problem (1.5)–(1.10). A function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b(x)\in \text {Lip}([0,\infty ))$$\end{document} is called a generating curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} of the cone surface as defined in (1.7), and a vector function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U=(u,v)^\top \in (\text {BV}_{\textrm{loc}}\cap \text {L}^{\infty })(\Omega )$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} defined in (1.6) is called an entropy solution of (1.5)–(1.10) if they satisfy the following conditions:

• (i)For any test function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \in \text {C}_{0}^1({\mathbb {R}}^2;{\mathbb {R}})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi \in \text {C}_{0}^1(\Omega ;{\mathbb {R}})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\iint _{\Omega }\big (\rho u\phi _x+\rho v\phi _y-\dfrac{\rho v}{y}\phi \big )\,{\textrm{d}}x {\textrm{d}}y +\int _{-\infty }^{0}\phi (0,y)\rho _{\infty }u_{\infty }\,{\textrm{d}}y=0,\\&\iint _{\Omega }\left( v\psi _x-u\psi _y\right) {\textrm{d}}x {\textrm{d}}y=0, \end{aligned} \end{aligned}$$\end{document}

• (ii)For any convex entropy pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathcal {E}},{\mathcal {Q}})$$\end{document} with respect to W of (1.5), i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^2{\mathcal {E}}(W)\geqq 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla {\mathcal {Q}}(W)=\nabla {\mathcal {E}}(W)\nabla H(U(W))$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\iint _{\Omega }\big ({\mathcal {E}}(W(U))\varphi _x+{\mathcal {Q}}(W(U))\varphi _y +\nabla _W{\mathcal {E}}(W(U))E(U,y)\varphi \big )\,{\textrm{d}}x {\textrm{d}}y\\&\quad \quad +\int _{-\infty }^{0}{\mathcal {E}}(W(U_{\infty }))\varphi (0,y)\,{\textrm{d}}y\geqq 0 \end{aligned} \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in \text {C}_{0}^1(\Omega ;{\mathbb {R}})$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \geqq 0$$\end{document} .

Remark 1.1

For the potential flow, the Bernoulli law (1.2) gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{M^2}{2}+\frac{1}{\gamma -1}=\frac{B_\infty }{c^2} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\infty =\frac{u_\infty ^2}{2}+\frac{c_\infty ^2}{\gamma -1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^2=\gamma \rho ^{\gamma -1}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=\rho ^\gamma $$\end{document} . Then the assumptions on pressure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^b$$\end{document} can be reduced to the equivalent ones on the Mach number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_b$$\end{document} on the unknown boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} .

Main Theorem

(Existence and stability).  Let (A1)–(A2) hold, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\gamma <3$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0<p_{0}<p^*:=\Big ((\sqrt{\gamma +7}-\sqrt{\gamma -1}) \sqrt{\frac{\gamma -1}{16\gamma }} \Big )^\frac{2\gamma }{\gamma -1}. \end{aligned}$$\end{document}

Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} is sufficiently large and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document} is sufficiently small such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \text {T.V.}\,\{p^{b}\}=\varepsilon _p<\varepsilon _0. \end{aligned} \end{aligned}$$\end{document}

Then there exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0$$\end{document} and the system, such that the following statements hold:

• (i)Global existence: IBVP (1.4)–(1.10) determines a boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=b(x)=\int _{0}^{x}b'_+(t)\,{\textrm{d}}t$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_+\in $$\end{document} BV \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb {R}}_{+})$$\end{document} satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x>0}|b'_+(x)-b_0|<C\varepsilon _p \end{aligned}$$\end{document}

and admits a global entropy solution U(x, y) with bounded total variation:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x>0}\text {T.V.}\,\{U(x,y)\, :\, -\infty<y<b(x)\}<\infty \end{aligned}$$\end{document}

in the sense of Definition 1.1. Moreover, the solution contains a strong leading shock-front \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=\chi (x)=\int _{0}^xs(t){\textrm{d}}t$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in $$\end{document} BV \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathbb {R}}_{+})$$\end{document} satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x>0} |s(x)-s_0|<C\varepsilon _p, \end{aligned}$$\end{document}

and the solution between the leading shock-front and the cone surface satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x>0}\text {T.V.}\{U(x,y)\, :\, \chi (x)<y<b(x)\}<C(\varepsilon _p + b_0 - s_0) . \end{aligned}$$\end{document}

Here above, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_0$$\end{document} denotes the slope of the corresponding straight-sided shock-front and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{0}$$\end{document} is the slope of the generating curve of the straight-sided cone surface.

• (ii)Asymptotic behavior: For the entropy solution U(x, y),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{x\rightarrow \infty }\sup \left\{ \big |U(x,y)-{\tilde{U}} (\sigma ;s_{\infty },G(s_{\infty }))\big |\,:\,\, \chi (x)<y<b(x)\right\} =0 \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(\sigma ;s_{\infty },G(s_{\infty }))$$\end{document} satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(s_{\infty };s_{\infty },G(s_{\infty }))=G(s_{\infty })$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&{\tilde{U}}(b_{\infty }';s_{\infty },G(s_{\infty }))\cdot (-b_{\infty }',1)=0,\\&\dfrac{1}{2}\big |{\tilde{U}}(b_{\infty }';s_{\infty },G(s_{\infty }))\big |^2 +\dfrac{\gamma (p^{b}_{\infty })^{\frac{\gamma -1}{\gamma }}}{\gamma -1} =\dfrac{1}{2}+\dfrac{\gamma p_{\infty }^{\frac{\gamma -1}{\gamma }}}{\gamma -1}, \end{aligned} \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} p^{b}_{\infty }=\lim _{x\rightarrow \infty }p^{b}(x),\quad \,\, s_{\infty }=\lim _{x\rightarrow \infty }s(x),\quad \,\, b'_{\infty }=\lim _{x\rightarrow \infty }b'_{+}(x), \end{aligned} \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(\sigma ;s,G(s))$$\end{document} is the state of the self-similar solution, and G(s) denotes the state connected to state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\infty }$$\end{document} by the strong leading shock-front of speed s.

During the last forty years, the shock stability problem has been studied for the perturbed cones with small perturbations of the straight-sided cone. For polytropic potential flow near the cone vertex, the local existence of piecewise smooth solutions was established in [15, 17] for both symmetrically perturbed cone and pointed body, respectively. Lien-Liu in [38] first analyzed the global existence of weak solutions via a modified Glimm scheme for the uniform supersonic isentropic Euler flow past over a piecewise straight-side cone, provided that the cone has a small opening angle (the initial strength of the shock-front is relatively weak) and the Mach number of the incoming flow is sufficiently large. Later on, Wang-Zhang considered in [48] for supersonic potential flow for the adiabatic exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (1,3)$$\end{document} over a symmetric Lipschitz cone with an arbitrary opening angle less than the critical angle and constructed global weak solutions that are small perturbations of the self-similar solution, given that the total variation of the slopes of the perturbed generating curves of the cone is sufficiently small and the Mach number of the incoming flow is sufficiently large. In addition, for the isothermal flows (i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1$$\end{document} ), Chen-Kuang-Zhang in [10] made full use of delicate expansions up to second-order as the Mach number of the incoming flow goes to infinity and provided a complete proof of the global existence and asymptotic behavior of conical shock-front solutions in BV when the isothermal flow passes through the Lipschitz perturbed cones that are small perturbations of the straight-sided one.

When the surface of the perturbed cone is smooth, using the weighted energy methods, Chen-Xin-Yin established the global existence of piecewise smooth solutions in [19]. They considered a 3-D axisymmetric potential flow past a symmetrically perturbed cone under the assumption that the attached angle is sufficiently small and the Mach number of the incoming flow is sufficiently large. This result was also extended to the M-D potential flow case; see [32] for more details. Under a certain boundary condition on the cone surface, the global existence of the M-D conical shock solutions was obtained in [49] when the uniform supersonic incoming flow with large Mach number passes a generally curved sharp cone. Meanwhile, using a delicate expansion of the background solution, Cui-Yin established the global existence and stability of a steady conical shock wave in [21, 22] for the symmetrically perturbed supersonic flow past an infinitely long cone whose vertex angle is less than the critical angle. More recently, by constructing new background solutions that allow the speeds of the incoming flows to approach the limit speed, the global existence of steady symmetrically conical shock solutions was established in Hu-Zhang [29] when a supersonic incoming potential flow hits a symmetrically perturbed cone with an opening angle less than the critical angle. We also remark that some pivotal results have been obtained on the stability of M-D transonic shocks under symmetric perturbations of the straight-sided cones or the straight-sided wedges, as well as on Radon measure solutions for steady compressible Euler equations of hypersonic-limit conical flows; see [6–8, 42, 50] and the references cited therein.

Corresponding to these shock stability problems, two types of inverse problems have been considered. One type is for the problems of determining the shape of the wedge in the planar steady supersonic flow for the given location of the leading shock front. This kind of inverse problems and the related inverse piston problems have been considered by Li-Wang in [34–37, 46, 47], where the leading shock-front is assumed to be smooth and the characteristic method is applied to find the piecewise smooth solution with the leading shock as its only discontinuity; see also [33]. The other one is for the problems of determining the shape of a wedge or a cone with given pressure distribution on it in the planar steady supersonic flow (cf. [41]) or axisymmetric conical steady supersonic flow. Though various numerical methods and the linearized method have been proposed to deal with this type of problems, there seems no rigorous result on the existence of solutions to such inverse problems for steady supersonic flow past a cone.

In this paper, we develop a modified Glimm scheme to establish the global existence and the asymptotic behavior of conical shock-front solutions of the inverse problem in BV in the flow direction, when the isentropic flow passes through the cones with given pressure distributions on their surfaces, which are small perturbations of a constant pressure less than the critical value. Mathematically, our problem can be formulated as a free boundary problem governed by 2-D steady isentropic irrotational Euler equations with geometric structure.

There are two main difficulties in solving this problem: one of them is the singularity generated by the geometric source term, and the other is that, compared to the shock stability problem for supersonic flows past a cone, the generating curve of the cone is unknown. For supersonic flows past an axisymmetric cone with the given generating curve, a modified Glimm scheme developed by Lien-Liu in [38] is used to construct approximate solutions (see also [10, 48]). In the previous construction, in order to incorporate with the geometric source term and the boundary condition on the approximate generating curve, the center \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_0,0)$$\end{document} of the self-similar variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =\frac{x-x_0}{y}$$\end{document} is defined to be the intersection of the x-axis and the line on which the current approximate generating curve (a line segment of a polyline) lies, and the center is changed according to the random choice at each step when the ordinary differential equations (2.3) are solved. As a result, the approximate solution on the approximate generating curve is a piecewise constant vector-valued function that satisfies the boundary condition everywhere. However, in the inverse problem under consideration in this paper, the generating curve of the cone is to be determined, apriori unknown, so that the approach in Lien-Liu (cf. [38]) could not apply directly.

To overcome the new difficulties, we first fix the center of the self-similar variable to be the origin when solving the differential equations (2.3) and then develop a modified Glimm scheme to construct approximate solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }(x,y)$$\end{document} via the self-similar solutions as building blocks in order to incorporate the geometric source term. In our construction, the grid points are fixed at the beginning, which are the intersections of lines \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=x_h$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}$$\end{document} , and the rays issuing from the origin (the vertex point of the cone). Consequently, this construction allows us to find a new term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{b}(h)$$\end{document} to control the increasing part of the Glimm type functional near the approximate boundary (see Lemma 4.6), while it brings us an extra error so that the boundary conditions on the approximate boundary are no longer satisfied everywhere, but are satisfied at the initial point of each approximate boundary at each step. Nevertheless, in Proposition 6.3, we are able to prove that this error goes to zero as the grid size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} tends to zero.

Furthermore, we make careful asymptotic expansions of the self-similar solutions with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }^{-1}$$\end{document} . We then make full use of the asymptotic expansion analysis of the background solutions with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }^{-1}$$\end{document} to calculate the reflection coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{r,1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{w,2}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_s$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{w,2}$$\end{document} of the weak waves reflected from both the boundary and the strong leading shock, and of the self-similar solutions reflected from the strong leading shock to prove that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }\big (|K_{r,1}||K_{w,2}|+|K_{r,1}||K_s||\mu _{w,2}|\big )<1. \end{aligned}$$\end{document}

Based on this, we choose some appropriate weights, independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} , in the construction of the Glimm-type functional and show that the functional is monotonically decreasing. Then the convergence of the approximate solutions is followed by the standard approach for the Glimm-type scheme as in [24, 31]; see also [4, 14, 18, 23, 44]. Finally, the existence of entropy solutions and the asymptotic behavior of the entropy solutions are also proven.

The remaining part of this paper is organized as follows: In Section 2, we give some preliminaries of the homogeneous system (1.1) and then study Riemann-type problems in several cases and self-similar solutions of the unperturbed conic flow. Also, we calculate the limit states of related quantities as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty } \rightarrow \infty $$\end{document} . In Section 3, we construct a family of approximate solutions via a modified Glimm scheme. In Section 4, we establish some essential interaction estimates in a small neighborhood in the limit state. Then, in Section 5, we define the Glimm-type functional and show the monotonicity of the Glimm-type functional and, in Section 6, we prove that there exists a subsequence of approximate solutions converging to the entropy solution. Finally, in Section 7, we give the asymptotic behavior of the entropy solution which, together with the existence theory, leads to our main theorem.

Riemann Problems and Self-Similar Solutions of the Unperturbed Conic Flow

Regarding x as the time variable, the simplified system of (1.1):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&(\rho u)_x+(\rho v)_y=0, \\&v_x-u_y=0, \end{aligned} \right. \end{aligned}$$\end{document}

is strictly hyperbolic with two distinctive eigenvalues:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{i}=\frac{uv+(-1)^ic\sqrt{u^2+v^2-c^2}}{u^2-c^2},\qquad i=1,\ 2, \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u>c_*$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^2+v^2<q_*^2$$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_*=\sqrt{\frac{\gamma -1}{\gamma +1}+\frac{2c_{\infty }^2}{\gamma +1}}, \quad \, q_*=\sqrt{1+\frac{2c_{\infty }^2}{\gamma +1}}. \end{aligned}$$\end{document}

Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q:=\sqrt{u^2+v^2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta :=\arctan \frac{v}{u}$$\end{document} . Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _{i}=\tan (\theta +(-1)^i\theta _{m}),\qquad i=1, 2, \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _{m}:=\arctan (\frac{c}{\sqrt{q^2-c^2}}) \end{aligned}$$\end{document}

is the Mach angel. A direct computation indicates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{m}\in (0,\frac{\pi }{2})$$\end{document} .

Next, we introduce the following lemma, whose proof can be found in [48]:

Lemma 2.1

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u>c_*$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<q_*$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( -\lambda _{i},1\right) \cdot (\frac{\partial \lambda _{i}}{\partial u},\frac{\partial \lambda _{i}}{\partial v}) = \frac{\gamma +1}{2\sqrt{q^2-c^2}}\sec ^3(\theta +(-1)^i\theta _{m}),\qquad i=1, 2. \end{aligned}$$\end{document}

Then, setting

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} r_{i}(U)=\frac{(-\lambda _{i}(U),1)}{(-\lambda _{i}(U),1)\cdot \nabla \lambda _{i}(U)},\qquad i=1, 2, \end{aligned}$$\end{document}

we see that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{i}(U)\cdot \nabla \lambda _{i}(U)=1$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1, 2$$\end{document} .

Denote the supersonic part of the shock polar by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S((u_{\infty },0))=\big \{({\bar{u}},{\bar{v}})\,:\, {\bar{c}}^2< {\bar{u}}^2+{\bar{v}}^2\leqq 1\big \}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\bar{u}},{\bar{v}})$$\end{document} satisfies the Rankine–Hugoniot condition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&{\bar{\rho }}({\bar{u}}s-{\bar{v}})=\rho _{\infty }s,\\&{\bar{u}}+{\bar{v}}s=1, \end{aligned} \right. \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\bar{u}}^2+{\bar{v}}^2}{2}+\frac{\gamma {\bar{\rho }}^{\gamma -1}}{\gamma -1} =\frac{1}{2}+\frac{c_{\infty }^2}{\gamma -1}$$\end{document} . Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S_{1}^-((u_{\infty },0))=\big \{({\bar{u}},{\bar{v}}):\, ({\bar{u}},{\bar{v}})\in S((u_{\infty },0)),\,{\bar{v}}<0\big \} \end{aligned}$$\end{document}

be the part of shock polar corresponding to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{1}$$\end{document} –characteristic field. Similarly to [30, 51, 52], we can parameterize the shock polar \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{1}^-((u_{\infty },0))$$\end{document} by a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {C}^2$$\end{document} –function:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G:s\mapsto G(s;U_{\infty }) \qquad \,\,\text{ with } U_{\infty }=(1,0). \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(s;U_{\infty })$$\end{document} is a supersonic state connected with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\infty }$$\end{document} by a shock of speed s. For simplicity, we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(s;U_{\infty })$$\end{document} as G(s) and use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{u}}(s)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{v}}(s)$$\end{document} to denote the components of G(s), that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(s)=({\bar{u}}(s),{\bar{v}}(s))^\top $$\end{document} . Then we have the following property for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{1}^-((u_{\infty },0))$$\end{document} (cf. [16]):

Lemma 2.2

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s<\lambda _{1}(U_{\infty })$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{\bar{\rho }}(s)$$\end{document} is a strictly monotonically decreasing function of s, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{u}}(s)$$\end{document} is a strictly monotonically increasing function of s.

As in [20], let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =\frac{y}{x}$$\end{document} . Then the equations in (1.1) become

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&\big (1-\frac{u^2}{c^2}\big )\sigma ^2 u_{\sigma } -\frac{2uv}{c^2}\sigma ^2 v_{\sigma } -\big (1-\frac{v^2}{c^2}\big )\sigma v_{\sigma }- v=0, \\&u_{\sigma }+\sigma v_{\sigma }=0. \end{aligned} \right. \end{aligned}$$\end{document}

or, equivalently,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&u_\sigma =\frac{c^2v}{(1+\sigma ^2)c^2-(v-\sigma u)^2},\\&v_\sigma =-\frac{c^2v}{\sigma \big ((1+\sigma ^2)c^2-(v-\sigma u)^2\big )},\\&\rho _\sigma =\frac{\rho v(v-\sigma u)}{\sigma \big ((1+\sigma ^2)c^2-(v-\sigma u)^2\big )}. \end{aligned} \right. \end{aligned}$$\end{document}

Given a constant state \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\bar{u}},{\bar{v}})=G(s)$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{1}^-((u_{\infty },0))$$\end{document} , there exists a local solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(\sigma ;s,G(s))=({\tilde{u}}(\sigma ;s),{\tilde{v}}(\sigma ;s))$$\end{document} of system (2.3) with initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ({\tilde{u}}(s;s),\,{\tilde{v}}(s;s))=({\bar{u}},\,{\bar{v}}). \end{aligned}$$\end{document}

This solution can be extended to an end-point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{u}}(\sigma _e;s),\,{\tilde{v}}(\sigma _e;s))$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\tilde{v}}(\sigma _e;s)}{{\tilde{u}}(\sigma _e;s)}=\sigma _e$$\end{document} . As \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\bar{u}},{\bar{v}})$$\end{document} varies on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{1}^-((u_{\infty },0))$$\end{document} , the collection of these end-states forms an apple curve through \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\infty }$$\end{document} (Fig. 3); see [20]. For these solutions, we have following properties, whose proof can be found in [48]:

Lemma 2.3

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{u}}(s;s)>{\tilde{c}}(s;s)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (s,\sigma _e)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{u}}(\sigma ;s)\sigma -{\tilde{v}}(\sigma ;s)<0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\partial {\tilde{u}}}{\partial \sigma }<0,\qquad \frac{\partial {\tilde{v}}}{\partial \sigma }<0,\\&{\tilde{c}}(\sigma ;s)-\frac{{\tilde{v}}(\sigma ;s)-\sigma {\tilde{u}}(\sigma ;s)}{\sqrt{1+\sigma ^2}}>{\tilde{c}}(s;s)-\frac{{\tilde{v}}(s;s)-s{\tilde{u}}(s;s)}{\sqrt{1+s^2}}>0, \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\tilde{u}}^2+{\tilde{v}}^2}{2}+\frac{{\tilde{c}}^2}{\gamma -1}= \frac{1}{2}+\frac{c_{\infty }^2}{\gamma -1}$$\end{document} .

Thus, we obtain the following estimate of the self-similar solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{u}}(\sigma ;s),\,{\tilde{v}}(\sigma ;s))$$\end{document} :

Lemma 2.4

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{u}}(s;s)>{\tilde{c}}(s;s)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (s,\sigma _e)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{1}{1+s^2}< {\tilde{u}}(\sigma ;s)<{\tilde{u}}(s;s),\\&{\tilde{c}}(s;s)<{\tilde{c}}(\sigma ;s)<{\tilde{c}}(\sigma _e;s)<\sqrt{\frac{(\gamma -1)s^2}{2(1+s^2)}+\frac{1}{M_{\infty }^2}}. \end{aligned}$$\end{document}

Fig. 3. Apple curve

To obtain the asymptotic expansion of the self-similar solution, we need the following properties of the shock polar:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p^*=\Big (\big (\sqrt{\gamma +7}-\sqrt{\gamma -1}\big ) \sqrt{\frac{\gamma -1}{16\gamma }} \Big )^\frac{2\gamma }{\gamma -1}. \end{aligned}$$\end{document}

Lemma 2.5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\gamma <3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}\in (0,p^*)$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} large enough, the equations:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&\rho _{0}(u^{\sharp }s^{\sharp }-v^{\sharp })=\rho _{\infty }s^{\sharp },\\&u^{\sharp }+v^{\sharp } s^{\sharp }=1,\\&\frac{(u^{\sharp })^2+(v^{\sharp })^2}{2}+\frac{c_{0}^2}{\gamma -1}=\frac{1}{2}+\frac{c_{\infty }^2}{\gamma -1} \end{aligned} \right. \end{aligned}$$\end{document}

have a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u^{\sharp },v^{\sharp },s^{\sharp })$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^{\sharp }<0$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{\infty }=p_{\infty }^{\frac{1}{\gamma }}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{0}=p_{0}^{\frac{1}{\gamma }}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{0}=\sqrt{\gamma } p_{0}^{\frac{\gamma -1}{2\gamma }}$$\end{document} , such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \lim _{M_{\infty }\rightarrow \infty }u^{\sharp }&=\lim _{M_{\infty }\rightarrow \infty }u_{a}=1-\frac{2c_{0}^2}{\gamma -1},\\ \lim _{M_{\infty }\rightarrow \infty }v^{\sharp }&=\lim _{M_{\infty }\rightarrow \infty }v_{a} =-\sqrt{\frac{2c_{0}^2}{\gamma -1-2c_{0}^2}}\Big (1-\frac{2c_{0}^2}{\gamma -1}\Big ),\\ \lim _{M_{\infty }\rightarrow \infty } c_{a}^2&= c_{0}^2, \end{aligned} \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{a}:=\frac{1}{1+(s^{\sharp })^2},\quad v_{a}:=\frac{s^{\sharp }}{1+(s^{\sharp })^2}, \quad c_{a}:=\sqrt{\frac{(\gamma -1)(s^{\sharp })^2}{2(1+(s^{\sharp })^2)}+\frac{1}{M_{\infty }^2}}. \end{aligned}$$\end{document}

Proof

From the first two equations of (2.6), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u^{\sharp }=\frac{\rho _{0}+\rho _{\infty }(s^{\sharp })^2}{\rho _{0}(1+s^2)}, \qquad v^{\sharp }=\frac{(\rho _{0}-\rho _{\infty })s^{\sharp }}{\rho _{0}\big (1+(s^{\sharp })^2\big )}. \end{aligned}$$\end{document}

With the help of the third equation of (2.6), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \bigg ( \frac{\rho _{0}+\rho _{\infty }(s^{\sharp })^2}{\rho _{0}\big (1+(s^{\sharp })^2\big )}\bigg )^2 +\bigg (\frac{(\rho _{0}-\rho _{\infty })s^{\sharp }}{\rho _{0}\big (1+(s^{\sharp })^2\big )} \bigg )^2 =1+\frac{2(c_{\infty }^2-c_{0}^2)}{\gamma -1}, \end{aligned}$$\end{document}

which gives that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (s^{\sharp })^2&=\frac{2(c_{0}^2-M_{\infty }^{-2})\rho _{0}^2}{(\gamma -1)\big (\rho _{0}^2-(\gamma M_{\infty }^2)^{-\frac{2}{\gamma -1}}\big )-2(c_{0}^2-M_{\infty }^{-2})\rho _{0}^2}\\&=\frac{2c_{0}^2}{\gamma -1-2c_{0}^2}-\frac{2(\gamma -1)}{(\gamma -1-2c_{0}^2)^2}\, M_{\infty }^{-2} +O(M_{\infty }^{-\frac{4}{\gamma -1}})\qquad \text {as } M_{\infty }\rightarrow \infty . \end{aligned}$$\end{document}

Therefore, noting that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^{\sharp }<0$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s^{\sharp }&=-\sqrt{\frac{2c_{0}^2}{\gamma -1-2c_{0}^2}} \bigg (1-\frac{\gamma -1}{2c_{0}^2\left( \gamma -1-2c_{0}^2\right) }\, M_{\infty }^{-2}\bigg ) +O(M_{\infty }^{-\frac{4}{\gamma -1}}) \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }\rightarrow \infty $$\end{document} . Substituting the above expansion into (2.8)–(2.9) yields (2.7). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 2.6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\gamma <3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}\in (0,p^*)$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} large enough, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{d}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \rho _{0}^{\gamma -1}=\frac{\rho _{d}^{\gamma +1}-\rho _{\infty }^{\gamma +1}}{\rho _{d}^2-\rho _{\infty }^2}, \end{aligned}$$\end{document}

and the equations:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&\rho _{d}(u_{d}s_{d}-v_{d})=\rho _{\infty }s_{d},\\&u_{d}+v_{d}s_{d}=1,\\&\frac{u_{d}^2+v_{d}^2}{2}+\frac{c_{d}^2}{\gamma -1}=\frac{1}{2}+\frac{c_{\infty }^2}{\gamma -1}, \end{aligned} \right. \end{aligned}$$\end{document}

have a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_{d},v_{d},s_{d})$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{d}<0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{d}=\sqrt{\gamma \rho _{d}^{\gamma -1}}$$\end{document} . Moreover, for

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u^{\flat }:=\frac{1}{1+s_{d}^2},\quad \, v^{\flat }:=\frac{s_{d}}{1+s_{d}^2}, \end{aligned}$$\end{document}

we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \lim _{M_{\infty }\rightarrow \infty } c_{d}^2&=c_{0}^2,\\ \lim _{M_{\infty }\rightarrow \infty }u^{\flat }&=\lim _{M_{\infty }\rightarrow \infty }u_{d} =1-\frac{2c_{0}^2}{\gamma -1},\\ \lim _{M_{\infty }\rightarrow \infty } v^{\flat }&= \lim _{M_{\infty }\rightarrow \infty } v_{d} =-\sqrt{\frac{2c_{0}^2}{\gamma -1-2c_{0}^2}}\,\Big (1-\frac{2c_{0}^2}{\gamma -1}\Big ). \end{aligned} \end{aligned}$$\end{document}

Proof

For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{0}$$\end{document} , it is direct to find \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _{d}$$\end{document} such that (2.10) holds. Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{d}\in (0,p^*)$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} is large enough. By Lemma 2.5, we obtain the unique solution of (2.11):

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s_{d}=\frac{2(c_{0}^2-c_{\infty }^2)}{\gamma -1-2(c_{0}^2-c_{\infty }^2)},\quad \, u_{d} =\frac{\rho _{d}+\rho _{\infty }s_{d}^2}{\rho _{d}(1+s_{d}^2)}, \quad \, v_{d}=\frac{(\rho _{d}-\rho _{\infty })s_{d}}{\rho _{d}(1+s_{d}^2)}. \end{aligned}$$\end{document}

Then we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s_{d}&=-\sqrt{\frac{2c_{0}^2}{\gamma -1-2c_{0}^2}} \Big (1-\frac{\gamma -1}{2c_{0}^2(\gamma -1-2c_{0}^2)}\, M_{\infty }^{-2}\Big ) +O(M_{\infty }^{-4}) \end{aligned}$$\end{document}

as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }\rightarrow \infty $$\end{document} .

Substituting the above expansion into (2.12)–(2.14) yields (2.13). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Given that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}\in (0,p^{*})$$\end{document} , we now solve the problem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{0}<\sigma <b_{0}$$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \sigma ^2\big (1-\frac{{\tilde{u}}^2}{{\tilde{c}}^2}\big ) {\tilde{u}}_{\sigma } -\frac{2{\tilde{u}}{\tilde{v}}\sigma ^2}{c^2}{\tilde{v}}_{\sigma } - \big (1-\frac{{\tilde{v}}^2}{{\tilde{c}}^2}\big ) {\tilde{v}}_{\sigma }\sigma -{\tilde{v}}=0,\\ {\tilde{u}}_{\sigma }+\sigma {\tilde{v}}_{\sigma }=0, \end{array}\right. } \end{aligned}$$\end{document}

with the boundary conditions

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\,\,\,{\tilde{\rho }}({\tilde{u}}s_{0}-{\tilde{v}})=\rho _{\infty }s_{0},\quad {\tilde{u}}+{\tilde{v}}s_{0}=1 \qquad \,\, & \text{ for } \sigma =s_{0},\\&\,\,\,{\tilde{\rho }}=\rho _{0},\quad {\tilde{v}}=b_{0}{\tilde{u}} \qquad \,\, & \text{ for } \sigma =b_{0}, \end{aligned} \end{aligned}$$\end{document}

and define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{u}}(\sigma ;s_{0}),{\tilde{v}}(\sigma ;s_{0}))=(1,0)$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma <s_{0}$$\end{document} . Indeed, we have

Lemma 2.7

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\gamma <3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}\in (0,p^{*})$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }>K_{1}$$\end{document} , problem (2.15) has a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{u}}(\sigma ;s_{0}),\,{\tilde{v}}(\sigma ;s_{0}))$$\end{document} containing a supersonic conical shock-front issuing from the vertex. In addition,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }(\sigma , {\tilde{c}}^2(\sigma ;s_{0}))=(\tan \theta _{0}, c_{0}^2),\\&\lim _{M_{\infty }\rightarrow \infty }({\tilde{u}}(\sigma ;s_{0}),{\tilde{v}}(\sigma ;s_{0})) =(\cos ^2\theta _{0},\,\sin \theta _{0}\cos \theta _{0}), \end{aligned} \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }\frac{{\tilde{u}}(\sigma ;s_{0})}{{\tilde{c}}(\sigma ;s_{0})} =\frac{\gamma -1-2c_{0}^2}{(\gamma -1)c_{0}}>1,\qquad \cos (\theta _{0}\pm \theta _{m}^{0})>0, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{0}=-\arctan (\sqrt{\frac{2c_{0}^2}{\gamma -1-2c_{0}^2}})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{m}^{0}=\lim _{M_{\infty }\rightarrow \infty }\theta _{m}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in [s_{0},b_{0})$$\end{document} .

Proof

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}\in (0,p^{*})$$\end{document} , by the shooting method as in [20], problem (2.15) has a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{u}}(\sigma ;s_{0}),{\tilde{v}}(\sigma ;s_{0}))$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{u}}(s_{0};s_{0})>{\tilde{c}}(s_{0};s_{0})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{\rho }}(b_{0};s_{0})=p_{0}^{\frac{1}{\gamma }}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{v}}(b_{0};s_{0})=b_{0}{\tilde{u}}(b_{0};s_{0})$$\end{document} .

We then focus on the asymptotic expansions (2.17). Lemma 2.4 indicates that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{1}{1+s_{0}^2}<{\tilde{u}}(\sigma ;s_{0})\leqq {\tilde{u}}(s_{0};s_{0}),\\ &\qquad {\tilde{c}}(s_{0};s_{0})\leqq {\tilde{c}}(\sigma ;s_{0})<c_{0} <\sqrt{\frac{(\gamma -1)s_{0}^2}{2(1+s_{0}^2)}+\frac{1}{M_{\infty }^2}} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in [s_{0},b_{0})$$\end{document} . Meanwhile, it follows from (2.10) that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{c}}(s_{0};s_{0})>c_{d}$$\end{document} . Then, due to Lemma 2.2, we see that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^{\sharp }<{\tilde{u}}(s_{0};s_{0})<u_{d}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^{\sharp }<s_{0}<0$$\end{document} . Therefore, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_{d}&<{\tilde{c}}(s_{0};s_{0}) \leqq {\tilde{c}}(\sigma ;s_{0})<c,\\ u_{a}&=\frac{1}{1+(s^{\sharp })^2}<\frac{1}{1+s_{0}^2}< {\tilde{u}}(\sigma ;s_{0})\leqq {\tilde{u}}(s_{0};s_{0})<u_{d}. \end{aligned}$$\end{document}

From Lemma 2.5–2.6, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }{\tilde{c}}(\sigma ;s_{0})=c_{0}^2, \qquad \lim _{M_{\infty }\rightarrow \infty }{\tilde{u}}(\sigma ;s_{0})=1-\frac{2c_{0}^2}{\gamma -1}. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{v}}(\sigma ;s_{0})<0$$\end{document} , from the Bernoulli laws,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{{\tilde{u}}^2+{\tilde{v}}^2}{2}+\frac{{\tilde{c}}^2}{\gamma -1}= \frac{1}{2}+\frac{c_{\infty }^2}{\gamma -1},\quad \, \frac{(u^{\sharp })^2+(v^{\sharp })^2}{2}+\frac{c_{0}^2}{\gamma -1}=\frac{1}{2}+\frac{c_{\infty }^2}{\gamma -1}, \end{aligned}$$\end{document}

we conclude

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }{\tilde{v}}(\sigma ;s_{0})=&-\sqrt{\frac{2c_{0}^2}{\gamma -1-2c_{0}^2}}\, \Big (1-\frac{2c_{0}^2}{\gamma -1}\Big ). \end{aligned}$$\end{document}

Again, by Lemma 2.2, we know that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{{\tilde{v}}(s_{0};s_{0})}{{\tilde{u}}(s_{0};s_{0})}>b_{0}>\sigma \geqq s_{0}>s^{\sharp }. \end{aligned}$$\end{document}

Combining all the expansions obtained above, we obtain (2.17).

Furthermore, since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }\big ( {\tilde{u}}\sqrt{{\tilde{u}}^2+{\tilde{v}}^2-{\tilde{c}}^2}\big )^2 -( {\tilde{v}}{\tilde{c}})^2 =\lim _{M_{\infty }\rightarrow \infty }({\tilde{u}}^2-{\tilde{c}}^2)({\tilde{u}}^2+{\tilde{v}}^2)>0, \end{aligned}$$\end{document}

we conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \cos (\theta _{0}\pm \theta ^0_{ma}) =\lim _{M_{\infty }\rightarrow \infty }\cos (\theta \pm \theta _{m}) =\lim _{M_{\infty }\rightarrow \infty }\frac{ {\tilde{u}}\sqrt{{\tilde{u}}^2+{\tilde{v}}^2-{\tilde{c}}^2}\mp {\tilde{v}}{\tilde{c}}}{{\tilde{u}}^2+{\tilde{v}}^2}>0. \end{aligned}$$\end{document}

This completes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Next, for G(s), we have the following expansions, whose proof can be found in [48]:

Lemma 2.8

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(s)=({\bar{u}}(s),{\bar{v}}(s))^\top $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }({\bar{u}}(s_{0}),\,{\bar{v}}(s_{0})) =(\cos ^2\theta _{0},\,\cos \theta _{0}\sin \theta _{0}), \\&\lim _{M_{\infty }\rightarrow \infty }({\bar{u}}_s(s_{0}),\,{\bar{u}}_s(s_{0})) =(-\sin (2\theta _{0})\cos ^2\theta _{0},\,\cos (2\theta _{0})\cos ^2\theta _{0}), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{u}}_s(s)=\frac{{\textrm{d}}{\bar{u}}(s)}{{\textrm{d}} s}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{v}}_s(s)=\frac{{\textrm{d}}{\bar{v}}(s)}{{\textrm{d}} s}$$\end{document} .

Now, we introduce the elementary wave curves of system (2.1). We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {W}(p_{0},p_{\infty })$$\end{document} the curve formed by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(\sigma ;s_{0})=({\tilde{u}}(\sigma ;s_{0}),{\tilde{v}}(\sigma ;s_{0}))^\top $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{0}<\sigma <b_{0}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}$$\end{document} is the corresponding pressure of the state at the endpoint. As in [48] (also cf. [51, 52]), we parameterize the elementary i-wave curves for system (2.1) in a neighborhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {W}(p_{0},p_{\infty })$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} O_{r}\big (\mathbb {W}(p_{0},p_{\infty })\big )=\bigcup \limits _{s_{0}<\sigma<b_{0}}\big \{U:\ |U-{\tilde{U}}(\sigma ;s_{0})|<r\big \} \qquad \,\text{ for } \text{ some } r>0, \end{aligned}$$\end{document}

by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \alpha _{i}\mapsto \Phi _{i}(\alpha _{i};U) \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{i}\in \text {C}^2$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left. \frac{\partial \Phi _{i}}{\partial \alpha _{i}}\right| _{\alpha _{i}=0} =r_{i}(U) \qquad \text{ for } U\in O_{r}(\mathbb {W}(p_{0},p_{\infty })), \,\, i=1,2. \end{aligned}$$\end{document}

In the sequel, define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Phi (\alpha _{1},\alpha _{2};U)=\Phi _{2}(\alpha _{2};\Phi _{1}(\alpha _{1};U)). \end{aligned}$$\end{document}

Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(\sigma ;\sigma _{0},U_{l})$$\end{document} the solution to the ODE system (2.3) with initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\tilde{U}}|_{\sigma =\sigma _{0}}=U_{l} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}\in O_{r}\big (\mathbb {W}(p_{0},p_{\infty })\big )$$\end{document} . Then, as in [48], we have

Lemma 2.9

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}\in (0,p^{*})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty } \left. \frac{{\textrm{d}}{\tilde{U}}(\sigma ;\sigma _{0})}{{\textrm{d}}\sigma }\right| _{\{\sigma =\sigma _{0},\,U_{l}\in \mathbb {W}(p_{0},p_{\infty })\}} =\big (\sin \theta _{0}\cos ^3\theta _{0},-\cos ^4\theta _{0}\big )^\top . \end{aligned}$$\end{document}

With all the limits given above, we obtain the following lemma, which is essential in wave-interaction estimates:

Lemma 2.10

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}\in \mathbb {W}(p_{0},p_{\infty })$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }\det \big (r_{1}(U_{l}),r_{2}(U_{l})\big )=\dfrac{4\cos ^2(\theta _{0}+\theta _{m}^{0})\cos ^2(\theta _{0}-\theta _{m}^{0}) \cos ^2\theta _{0}\cos ^2\theta ^{0}_{m}\sin (2\theta _{m}^{0})}{(\gamma +1)^2},\\&\lim _{M_{\infty }\rightarrow \infty }\det \big (G_s(s_{0}),r_{1}\big (G(s_{0})\big )\big ) =-\dfrac{2 \cos ^2(\theta _{0}-\theta _{m}^{0})\cos \theta ^{0}_{m}\cos ^3\theta _{0}\sin (\theta _{0}+\theta _{m}^{0})}{\gamma +1},\\&\lim _{M_{\infty }\rightarrow \infty }\det \big (r_{2}\big (G(s_{0})\big ),G_s(s_{0})\big ) =\dfrac{2 \cos ^2(\theta _{0}+\theta _{m}^{0})\cos \theta ^{0}_{m}\cos ^3\theta _{0}\sin (\theta _{0}-\theta _{m}^{0})}{\gamma +1},\\&\lim _{M_{\infty }\rightarrow \infty }\det \Big (\frac{{\textrm{d}}{\tilde{U}}(\sigma ;\sigma _{0},G(s_{0}))}{{\textrm{d}}\sigma },G_s(s_{0})\Big ) =- \cos ^5\theta _{0}\sin \theta _{0},\\&\lim _{M_{\infty }\rightarrow \infty }\det \Big (r_{2}(U_{l}),\frac{{\textrm{d}}{\tilde{U}}(\sigma ;\sigma _{0},U_{l})}{{\textrm{d}}\sigma }\Big ) =\frac{2}{\gamma +1} \cos ^4\theta _{0}\cos ^2(\theta _{0}+\theta _m^{0})\cos \theta _m^{0}\sin \theta _m^{0},\\&\lim _{M_{\infty }\rightarrow \infty }\det \Big (r_{1}(U_{l}),\frac{{\textrm{d}}{\tilde{U}}(\sigma ;\sigma _{0},U_{l})}{{\textrm{d}}\sigma }\Big ) =-\frac{2}{\gamma +1} \cos ^4\theta _{0}\cos ^2(\theta _{0}-\theta _m^{0})\cos \theta _m^{0}\sin \theta _m^{0}. \end{aligned}$$\end{document}

Furthermore, we have the following propositions, which will be used in the construction of building blocks of our approximate solutions:

Proposition 2.1

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} sufficiently large, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{1}>0$$\end{document} such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{r}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}$$\end{document} lie in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O_{\varepsilon _{1}}(\mathbb {W}(p_{0},p_{\infty }))$$\end{document} , the Riemann problem (2.1) with the initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U|_{x={\bar{x}}}=\left\{ \begin{aligned}&U_{r}&\,\,\, \text {for }y>{\bar{y}},\\&U_{l}&\,\,\, \text {for }y<{\bar{y}}, \end{aligned} \right. \end{aligned}$$\end{document}

admits a unique admissible solution consisting of at most two elementary waves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{1}$$\end{document} for the 1-characteristic field and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{2}$$\end{document} for the 2-characteristic field. Moreover, states \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{r}$$\end{document} are connected by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{r}=\Phi (\alpha _{1},\alpha _{2};U_{l}). \end{aligned}$$\end{document}

Proposition 2.2

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} sufficiently large, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{2}>0$$\end{document} such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}\in O_{\varepsilon _{2}}( \mathbb {W}(p_{0},p_{\infty }))$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{2}\in O_{\varepsilon _{2}}(p_{0})$$\end{document} , there is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{1}$$\end{document} solving the equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dfrac{1}{2}|\Phi (\delta _{1},0;U_{l})|^2+\dfrac{\gamma }{\gamma -1}p_{2}^{\frac{\gamma -1}{\gamma }} =\dfrac{1}{2}|U_{l}|^2+\dfrac{\gamma }{\gamma -1}p_{1}^{\frac{\gamma -1}{\gamma }}. \end{aligned}$$\end{document}

Proof

From (2.20), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }\frac{1}{2}\, \left. \frac{\partial |\Phi (\delta _{1},0;U_{l})|^2}{\partial \delta _{1}}\right| _{\delta _{1}=0} =U_{l}\cdot r_{1}(U_{l})\ne 0. \end{aligned}$$\end{document}

By the implicit function theorem, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{1}$$\end{document} such that (2.20) holds, provided \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{2}$$\end{document} sufficiently small. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proposition 2.3

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} sufficiently large, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{3}>0$$\end{document} such that, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}=U_{\infty }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{r}\in O_{\varepsilon _{3}}( \mathbb {W}(p_{0},p_{\infty }))\cap O_{\varepsilon _{3}}(G(s_{0}))$$\end{document} , the Riemann problem (2.1) with initial data (2.19) admits a unique admissible solution that contains a strong 1-shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{1}$$\end{document} and a 2-weak wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}$$\end{document} of the 2-characteristic field. Moreover, states \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{r}$$\end{document} are connected by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{r}=\Phi _{2}(\beta _{2};G(s_{1};U_{l})). \end{aligned}$$\end{document}

Proof

It follows from (2.21) and Lemma 2.10 that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\lim _{M_{\infty }\rightarrow \infty } \det \Big ( \frac{\partial \Phi _{2}(\beta _{2};G(s_{1};U_{l}))}{\partial (s_{1},\beta _{2})}\Bigg | _{\{s_{1}=s_{0}, \beta _{2}=0\}}\Big ) \\ &\quad =-\lim _{M_{\infty }\rightarrow \infty }\det \big (r_{2}(G(s_{0}),G_s(s_{0}))\big )\ne 0. \end{aligned} \end{aligned}$$\end{document}

The existence of the solution of this Riemann problem is ensured by the implicit function theorem for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{3}$$\end{document} sufficiently small. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

To end this section, we introduce the following interaction estimate given by Glimm [24] for weak waves (see also [44, 48, 52]):

Lemma 2.11

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}\in \mathbb {W}(p_{0},p_{\infty })$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} satisfy

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Phi (\delta _{1},\delta _{2};U_{l})=\Phi (\beta _{1},\beta _{2};\Phi (\alpha _{1},\alpha _{2};U_{l})). \end{aligned}$$\end{document}

Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta =\alpha +\beta +O(1)Q^0(\alpha ,\beta ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q^0(\alpha ,\beta )=\sum \{|\alpha _{i}||\beta _{i}|:\alpha _{i}\text { and }\beta _{j}\text { approach}\}$$\end{document} , and O(1) depends continuously on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }<\infty $$\end{document} .

Approximate Solutions

In this section, we construct approximate solutions for system (1.4) with (1.8)–(1.9) by a modified Glimm scheme. Compared to the modified Glimm scheme developed in [10, 38, 48], in our construction, the grid points are fixed at the very beginning, which are independent of the approximate solution and the random choice.

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x>0$$\end{document} , there exist piecewise constant functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{b}_{\Delta x}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {T.V.}\,\{p^{b}_{\Delta x}(\cdot )\}\leqq \text {T.V.}\,\{p^{b}(\cdot )\}, \qquad \Vert p^{b}_{\Delta x}-p^{b}\Vert _{{{\textbf {L}}}^\infty }\leqq \varepsilon , \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p^{b}_{\Delta x}(x)={\left\{ \begin{array}{ll} p^{b}_{\Delta x,0}=p_{0} \,\,\,& \text { for }x\in [0,x_{0}),\\ p^{b}_{\Delta x,h+1} \,\,\,& \text { for }x\in [x_{h},x_{h+1})\text { and }h\in {\mathbb {N}}, \end{array}\right. } \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{b}_{\Delta x,h+1}$$\end{document} being constants on the corresponding intervals and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{h}=x_{0}+h\Delta x$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}$$\end{document} . Then, from Lemma 2.7, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{b}_{\Delta x,0}=p_{0}$$\end{document} , there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\tilde{u}}(\sigma ;s_{0}),{\tilde{v}}(\sigma ;s_{0})\big )$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{p}}(b_{0};s_{0})=p_{0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{v}}(b_{0};s_{0})=b_{0}{\tilde{u}}(b_{0};s_{0})$$\end{document} .

We now define the difference scheme. Choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =(\vartheta _{0},\vartheta _{1},\vartheta _{2},\dots ,\vartheta _{h},\dots )$$\end{document} randomly in [0, 1). For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< x< x_{0}$$\end{document} , let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b_{\Delta x,\vartheta }(x)=b_{0}x,\qquad \,\, \chi _{\Delta x,\vartheta }(x)=s_{0}x. \end{aligned}$$\end{document}

We denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta ,0}=\big \{(x,b_{\Delta x,\vartheta }(x)):\, 0\leqq x< x_{0}\big \}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x,\vartheta ,0}=\big \{(x,\chi _{\Delta x,\vartheta }(x)):\, 0\leqq x< x_{0}\big \}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,0}=\big \{(x,y):\, y<b_{\Delta x,\vartheta }(x), \, 0\leqq x< x_{0}\big \}$$\end{document} . In region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,0}$$\end{document} , we then define that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&U_{\Delta x,\vartheta }(x,y)\\&\quad = {\left\{ \begin{array}{ll} (u_{\Delta x,\vartheta }(x,y),v_{\Delta x,\vartheta }(x,y))^\top \triangleq ({\tilde{u}}(\sigma ;s_{0}),{\tilde{v}}(\sigma ;s_{0}))^\top \,\,\, & \text {for }\frac{y}{x}=\sigma \in (s_{0},b_{0}),\\ U_{\infty } \quad & \text {for }\frac{y}{x}=\sigma <s_{0}, \end{array}\right. } \end{aligned}$$\end{document}

and, on boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta ,0}$$\end{document} , we set that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{\Delta x,\vartheta }(x, b_{\Delta x,\vartheta }(x))&=U_{\Delta x,\vartheta }^b(x)=(u_{\Delta x,\vartheta }^b(x),v_{\Delta x,\vartheta }^b(x))^\top \\&\triangleq ({\tilde{u}}(b_{0};s_{0}),{\tilde{v}}(b_{0};s_{0}))^\top . \end{aligned}$$\end{document}

On \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=x_{h}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}$$\end{document} , the grid points are defined to be the intersections of line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=x_{h}$$\end{document} with the self-similar rays

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} y=\left( b_{0}+n\Delta \sigma \right) x\qquad \text{ for } n\in {\mathbb {Z}}. \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma >0$$\end{document} is chosen so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma >\dfrac{4\Delta x}{x_{0}}\max _{i=1,2}\{|\lambda _i(G(s_{0}))|\}$$\end{document} , and hence the numerical grids satisfy the usual Courant-Friedrichs-Lewy condition. Then we define the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} to be a piecewise smooth solution to the self-similar system (2.4), the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }^b$$\end{document} on the boundary, the approximate boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }=\big \{(x,y):\,y=b_{\Delta x,\vartheta }(x)\big \}$$\end{document} , and the numerical grids inductively in h, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=0,1,2,\cdots $$\end{document} .

Suppose that the approximate solution has been defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x<x_{h}$$\end{document} . The grid points on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=x_{h}$$\end{document} are denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_n(h)$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {Z}}$$\end{document} . Set that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} r_{h,n}=y_{n}(h)+\vartheta _{h}\big (y_{n+1}(h)-y_{n}(h)\big )\qquad \,\, \text{ for } n\in {\mathbb {Z}}. \end{aligned}$$\end{document}

Then the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }(x_{h},y)$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in (y_{n}(h), y_{n+1}(h))$$\end{document} is defined to be the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{self,\Delta x,\vartheta }(\sigma (x,y))$$\end{document} of (2.3) with the self-similar variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (x,y)=\frac{y}{x_{h}}$$\end{document} and with the initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma =\frac{r_{h,n}}{x_{h}}:\quad U_{self,\Delta x,\vartheta } =U_{\Delta x,\vartheta }(x_{h},r_{h,n})\triangleq U_{\Delta x,\vartheta }(x_{h}-,r_{h,n}+) \qquad \text{ for } n\in {\mathbb {Z}}. \end{aligned}$$\end{document}

For the discontinuities at the grid points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n}(h))$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {Z}}$$\end{document} , we solve the Riemann problems for (2.1) with the Riemann data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U|_{x=x_{h}}=\left\{ \begin{aligned}&U_{\Delta x,\vartheta }(x_{h},y_{n}(h)-)\qquad \text{ for } y<y_{n}(h), \\&U_{\Delta x,\vartheta }(x_{h},y_{n}(h)+)\qquad \text{ for } y>y_{n}(h), \end{aligned} \right. \end{aligned}$$\end{document}

and the solution consisting of rarefaction waves and shock waves has form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{Rie}(\eta )$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta =\dfrac{y-y_n(h)}{x-x_{h}}$$\end{document} . Setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{h,n+\frac{1}{2}}\triangleq \dfrac{1}{2x_{h}}\big (y_{n+1}(h+1)+y_{n}(h+1)\big )$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in {\mathbb {Z}}$$\end{document} , then, in the region

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega _{h+1,n}=\big \{(x,y)\, :\, x_{h}< x<x_{h+1}, \sigma _{h,n+\frac{1}{2}}>\sigma >\sigma _{h,n-\frac{1}{2}}\big \}, \end{aligned}$$\end{document}

along the ray

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \big \{(x,y)\,:\, \frac{y-y_n(h)}{x-x_{h}}=\eta ,\,x_{h}<x<x_{h+1}\big \}, \end{aligned}$$\end{document}

the approximate solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }(x,y)$$\end{document} is defined to be the solution: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{self,\Delta x,\vartheta }(\sigma (x,y))$$\end{document} of (2.3) with the self-similar variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (x,y)=\frac{y}{x}$$\end{document} and with the initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma =\frac{y_{n}}{x_{h}}:\quad U_{self,\Delta x,\vartheta }=U_{Rie}(\eta ). \end{aligned}$$\end{document}

The approximate boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }=\big \{(x,y):\,y=b_{\Delta x,\vartheta }(x)\big \}$$\end{document} is traced continuously; see [10, 38, 48]. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in (0,x_{0})$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{\Delta x,\vartheta }(x)=b_{0}x$$\end{document} . Suppose that the approximate solution is constructed for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x<x_{h}$$\end{document} and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_{n_{b,h}}<b_{\Delta x,\vartheta }(x_{h}-)<y_{n_{b,h}+1}$$\end{document} . We call interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_{n_{b,h}-1}<y<y_{n_{b,h}+1}$$\end{document} the boundary region at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=x_{h}$$\end{document} . In this boundary region, we first solve the self-similar problem (2.3) with the initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma =\frac{r_{h,n_{b}-1}}{x_{h}}:\quad U_{self}=U_{\Delta x,\vartheta }(x_{h}-,r_{h,n_{b}-1}+), \end{aligned}$$\end{document}

and with the self-similar variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (x_{h},y)=\frac{y}{x_{h}}$$\end{document} . We denote the solution by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{self}(\sigma (x_{h},y))$$\end{document} . Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{b}_{\Delta x,h+1}$$\end{document} , by Proposition 2.2, there is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{2}\big |\Phi (\beta _{1},0;U_{self}(\sigma (x_{h},b_{\Delta x,\vartheta }(x_{h}))))\big |^2 +\frac{\gamma }{\gamma -1}(p^{b}_{\Delta x,h+1})^{\frac{\gamma -1}{\gamma }} =\frac{1}{2}+\frac{\gamma }{\gamma -1}p_{\infty }^{\frac{\gamma -1}{\gamma }}. \end{aligned}$$\end{document}

Then we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{\Delta x,\vartheta }^b(x_{h})\triangleq \Phi (\beta _{1},0;U_{self}(\sigma (x_{h},b_{\Delta x,\vartheta }(x_{h})))), \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b_{\Delta x,\vartheta }(x) =b_{\Delta x,\vartheta }(x_{h}-)+\frac{v^b_{\Delta x,\vartheta }(x_{h})}{u^b_{\Delta x,\vartheta }(x_{h})}\,(x-x_{h}) \qquad \, \text{ for } x\in [x_{h},x_{h+1}). \end{aligned}$$\end{document}

Next, solve again the self-similar problem (2.3) with initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}U_{-}(\sigma (x_{h},b_{\Delta x,\vartheta }(x_{h})))=U_{\Delta x,\vartheta }^b(x_{h})\end{aligned}$$\end{document}

and with the self-similar variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (x_{h},y)=\frac{y}{x_{h}}$$\end{document} . Denote the solution by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{-}(\sigma (x_{h},y))$$\end{document} . We define the approximate solution in the boundary region as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{\Delta x,\vartheta }(x_{h},y)=U_{-}(\sigma (x_{h},y))\qquad \, \text{ for } x_{h}\leqq x<x_{h+1}. \end{aligned}$$\end{document}

The discontinuities at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{b,h}-1})$$\end{document} are resolved by the same methods as before.

The leading strong conical shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x,\vartheta }=\big \{(x,y):\,y=\chi _{\Delta x,\vartheta }(x)\big \}$$\end{document} next to the uniform upstream flow is also traced continuously; see [10, 38, 48]. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in (0,x_{0})$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\Delta x,\vartheta }(x)=s_{0}x$$\end{document} . Suppose that the approximate solution is constructed for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x<x_{h}$$\end{document} and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_{n_{\chi ,h}-1}<\chi _{\Delta x,\vartheta }(x_{h}-)<y_{n_{\chi ,h}}$$\end{document} . We call interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_{n_{\chi ,h}-1}<y<y_{n_{\chi ,h}+1}$$\end{document} the front region at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=x_{h}$$\end{document} . In this front region, we first solve the self-similar problem (2.3) with the initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma =\frac{r_{h,n_{\chi }}}{x_{h}}:\quad U_{self}=U_{\Delta x,\vartheta }(x_{h}-,r_{h,n_{\chi }}+), \end{aligned}$$\end{document}

and with the self-similar variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (x_{h},y)=\frac{y}{x_{h}}$$\end{document} . Denote the solution by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{self}(\sigma (x_{h},y))$$\end{document} . Then we solve the Riemann problem (2.1) with the initial data

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U(x_{h},y)=\left\{ \begin{aligned}&U_{\infty },\,\,\, & y<\chi _{\Delta x,\vartheta }(x_{h}-), \\&U_{self}(\sigma (x_{h},\chi _{\Delta x,\vartheta }(x_{h}))), \,\,\, & \chi _{\Delta x,\vartheta }(x_{h}-)<y<y_{n_{\chi ,h}+1}. \end{aligned} \right. \end{aligned}$$\end{document}

The solution U(x, y) contains a weak 2-wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}$$\end{document} and a relatively strong 1-shock wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{\Delta x,\vartheta }(h+1)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{self}(\sigma (x_{h},\chi _{\Delta x,\vartheta }(x_{h})))=\Phi (0,\beta _{2};G(s_{\Delta x,\vartheta }(h+1);U_{\infty })). \end{aligned}$$\end{document}

Then we define that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \chi _{\Delta x,\vartheta }(x)=\chi _{\Delta x,\vartheta }(x_{h}-)+s_{\Delta x,\vartheta }(h+1)(x-x_{h}) \qquad \text{ for } x\in [x_{h},x_{h+1}). \end{aligned}$$\end{document}

Next, solve again the self-similar problem (2.3) with initial data \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{+}(\sigma (x_{h},\chi _{\Delta x,\vartheta }(x_{h})))=G(s_{\Delta x,\vartheta }(h+1);U_{\infty })$$\end{document} and with the self-similar variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma (x_{h},y)=\frac{y}{x_{h}}$$\end{document} . Denote the solution by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{+}(\sigma (x_{h},y))$$\end{document} . We define the approximate solution in the front region as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{\Delta x,\vartheta }(x_h,y)=\left\{ \begin{aligned}&U_{\infty },\,\,\, & y<\chi _{\Delta x,\vartheta }(x_{h}), \\&U_{+}(\sigma (x_{h},y)),\,\,\, & \chi _{\Delta x,\vartheta }(x_{h})<y<y_{n_{\chi ,h}+1}. \end{aligned} \right. \end{aligned}$$\end{document}

The discontinuities at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{\chi ,h}})$$\end{document} are resolved by the same methods as before.

Riemann-Type Problems and Interaction Estimates

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h}=\{(x,y)\,:\,y<b_{\Delta x,\vartheta }, x_{h-1}\leqq x< x_{h}\}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_{+}$$\end{document} . In order to define the approximate solutions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta }\triangleq \bigcup _{k=0}^\infty \Omega _{\Delta x,\vartheta ,k}$$\end{document} , the approximate boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }\triangleq \bigcup _{k=0}^\infty \Gamma _{\Delta x,\vartheta ,k}$$\end{document} , and the approximate leading shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x,\vartheta }\triangleq \bigcup _{k=0}^\infty S_{\Delta x,\vartheta ,k}$$\end{document} , we need a uniform bound of them to ensure that all the Riemann problems and the differential equations (2.3) are solvable. To achieve this, the following formulas are used:

• (i)If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \text {C}^1({\mathbb {R}})$$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(t)-f(0)=t\int _{0}^1f_t(\mu t){\textrm{d}}\mu \qquad \,\, \text{ for } t\in {\mathbb {R}}. \end{aligned}$$\end{document}

• (ii)If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \text {C}^2({\mathbb {R}})$$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&f(s,t)-f(s,0)-f(0,t)+f(0,0)\\&\quad = st\int _{0}^1\int _{0}^1f_{st}(\mu s,\lambda t){\textrm{d}}\mu {\textrm{d}}\lambda \qquad \,\, \text{ for } (s,t)\in {\mathbb {R}}^2. \end{aligned} \end{aligned}$$\end{document}

From now on, we use Greek letters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} to denote the elementary waves in the approximate solution, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{i}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{i}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _{i}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _{i}$$\end{document} stand for the corresponding i-th components for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} . As in [14, 44, 48, 52], a curve I is called a mesh curve provided that I is a space-like curve and consists of the line segments joining the random points one by one in turn. I divides region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x, \vartheta }$$\end{document} into two parts: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^-$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^+$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^-$$\end{document} denotes the part containing line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=x_{0}$$\end{document} . For any two mesh curves I and J, we use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J>I$$\end{document} to represent that every mesh point of curve J is either on I or contained in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^+$$\end{document} . We say J is an immediate successor to I if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J>I$$\end{document} and every mesh point of J except one is on I in general but three when these points are near the approximate boundary or the approximate shock.

Assume now that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} has been defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcup _{k=0}^h\Omega _{\Delta x,\vartheta ,k}$$\end{document} and the following conditions are satisfied:

• \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {{\text {H}}_{1}\text{(h) }}$$\end{document} :  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{S_{\Delta x,\vartheta ,k}\}_{k=0}^{h}$$\end{document} forms an approximate strong shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x,\vartheta }|_{0\leqq x<x_{h}}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\Gamma _{\Delta x,\vartheta ,k}\}_{k=0}^{h}$$\end{document} forms an approximate boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }|_{0\leqq x<x_{h}}$$\end{document} , both of which emanate from the origin;
• \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {{\text {H}}_{2}\text{(h) }}$$\end{document} : In each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,k}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqq k\leqq h$$\end{document} , the strong 1-shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x, \vartheta , k}$$\end{document} divides \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,k}$$\end{document} into two parts: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,k}^-$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,k}^+$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,k}^+$$\end{document} is the part between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x, \vartheta , k}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x, \vartheta , k}$$\end{document} ;
• \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {{\text {H}}_{3}\text{(h) }}$$\end{document} : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }|_{\Omega _{\Delta x,\vartheta ,k}^-}=U_{\infty }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\,U_{\Delta x,\vartheta }|_{\Omega _{\Delta x,\vartheta ,k}^+} \in O_{\varepsilon _{0}}(G(s_{0}))\cap O_{\varepsilon _{0}}(\mathbb {W}(p_{0},p_{\infty }))$$\end{document} , and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{\Delta x,\vartheta }(x,b_{\Delta x, \vartheta }(x)-)=U_{\Delta x,\vartheta , k}^{b} \in O_{\varepsilon _{0}}(G(s_{0}))\cap O_{\varepsilon _{0}}( \mathbb {W}(p_{0},p_{\infty })) \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{k}\leqq x<x_{k+1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\leqq k\leqq h$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\varepsilon _{0}<\min \{\varepsilon _{j},j=1,2,3\}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{j}$$\end{document} are introduced in Propositions 2.1–2.3 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2,3$$\end{document} . Then we prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} can be defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h+1}$$\end{document} satisfying conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{H}}_{1}(\hbox {h}+1)$$\end{document} – \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{H}}_3(\hbox {h}+1)$$\end{document} . As in [24] (see also [10, 14, 44]), we consider a pair of the mesh curves (I, J) lying in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_{h-1}<x<x_{h+1}\}\cap \Omega _{\Delta x, \vartheta }$$\end{document} with J being an immediate successor of I.

Now, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} be the region between I and J, and let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_{\Delta x,\vartheta }\in O_{\varepsilon _{0}}(G(s_{0}))\cap O_{\varepsilon _{0}}(\mathbb {W}(p_{0},p_{\infty })). \end{aligned}$$\end{document}

Case 1

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} is between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x, \vartheta }$$\end{document} .  In this case, we consider the interactions between weak waves. From the construction of the approximate solutions, the waves entering \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n-1}(h-1))$$\end{document} and from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n}(h-1))$$\end{document} are denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =(\alpha _{1},\alpha _{2})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =(\beta _{1},\beta _{2})$$\end{document} , respectively. We denote that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma _{0}&=\frac{r_{h-1,n}}{x_{h-1}},\&{\bar{\sigma }}_{0}&=\frac{r_{h-1,n-1}}{x_{h-1}},\&{\hat{\sigma }}_{0}&=\frac{r_{h-1,n-2}}{x_{h-1}},\\ \sigma _{1}&=\frac{y_{n}(h-1)}{x_{h-1}}=\frac{y_{n}(h)}{x_{h}},\&\sigma _{2}&=\frac{y_{n-1}(h-1)}{x_{h-1}}=\frac{y_{n-1}(h)}{x_{h}}, \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} U_{1}&=U_{\Delta x,\vartheta }(x_{h-1}-,r_{h-1,n}+),&U_{2}&=U_{\Delta x,\vartheta }(x_{h-1}-,r_{h-1,n-1}+),\\ U_3&=U_{\Delta x,\vartheta }(x_{h-1}-,r_{h-1,n-2}+).&\end{aligned} \end{aligned}$$\end{document}

Fig. 4. Interaction between weak waves

Case 1.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =(\delta _{1},\delta _{2})$$\end{document} be the waves issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n-1}(h))$$\end{document} ; see Fig. 4. Then we need to solve the following equations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =(\delta _{1},\delta _{2})$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (\delta _{1},\delta _{2};U_{l})) =\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (\alpha _{1},\alpha _{2};U_{l}))), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}={\tilde{U}}(\sigma _{2};{\hat{\sigma }}_{0},U_3)$$\end{document} .

Lemma 4.1

Equation (4.3) has a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =(\delta _{1},\delta _{2})$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta _{1}=\alpha _{1}+\beta _{1}+O(1)Q(\Lambda ),\qquad \delta _{2}=\alpha _{2}+O(1)Q(\Lambda ), \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q(\Lambda )=Q^0(\Lambda )+Q^1(\Lambda ) \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q^0(\Lambda )=\sum \big \{|\alpha _{j}||\beta _k|\,:\, \alpha _{j}\text { and }\beta _k\text { approach}\big \},\qquad Q^1(\Lambda )=|\beta _{1}||\Delta \sigma |, \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma =\sigma _{1}-\sigma _{2}$$\end{document} , and O(1) depends continuously on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} but independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , \beta , \Delta \sigma )$$\end{document} .

Proof

Lemma 2.10 yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty } \det \bigg (\left. \frac{\partial \Phi (\delta _{1},\delta _{2};U_{l})}{\partial (\delta _{1},\delta _{2})}\right| _{\{\delta _{1}=\delta _{2}=0,\, U_{l}\in \mathbb {W}(p_{0,p_{\infty }})\}}\bigg )\\&\quad =\dfrac{4\cos ^2(\theta _{0}+\theta _{m}^0)\cos ^2(\theta _{0}-\theta _{m}^0)\cos ^2\theta _{0}\cos ^2\theta ^0_{m}\sin (2\theta _{m}^0)}{(\gamma +1)^2}. \end{aligned}$$\end{document}

Then, by the implicit function theorem, system (4.3) has a unique \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {C}^2$$\end{document} –solution:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta =\delta (\alpha ,\beta ,\Delta \sigma ;U_{l}) \end{aligned}$$\end{document}

in a neighborhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha ,\beta ,\Delta \sigma ,U_{l})=(0,0,0,G(s_0))$$\end{document} . Due to (4.2), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta _{i}(\alpha ,\beta ,\Delta \sigma ;U_{l})&=\delta _{i}(\alpha ,0,\Delta \sigma ;U_{l})\\ &\quad +\delta _{i}(\alpha ,\beta ,0;U_{l}) -\delta _{i}(\alpha ,0,0;U_{l})+O(1)|\beta ||\Delta \sigma |\\&=\alpha _{i}+\beta _{i}+O(1)Q^0(\Lambda )+O(1)|\beta ||\Delta \sigma | \qquad \text{ for } i=1,2, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{2}=0$$\end{document} . Then the proof is complete. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Case 1.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =(\delta _{1},\delta _{2})$$\end{document} be the waves issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n}(h))$$\end{document} ; see Fig. 4. Then we need to solve the following equations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =(\delta _{1},\delta _{2})$$\end{document} :

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Phi (\delta _{1},\delta _{2};{\tilde{U}}(\sigma _{1};\sigma _{2},U_{l})) =\Phi (\beta _{1},\beta _{2};{\tilde{U}}(\sigma _{1},\sigma _{2};\Phi (0,\alpha _{2};U_{l}))), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}$$\end{document} satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}({\bar{\sigma }}_{0};\sigma _{1},\Phi (0,\alpha _{2};U_{l}))=U_{2}$$\end{document} . Similarly, we have

Lemma 4.2

Equation (4.4) has a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =(\delta _{1},\delta _{2})$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta _{1}=\beta _{1}+O(1)Q(\Lambda ),\qquad \delta _{2}=\alpha _{2}+\beta _{2}+O(1)Q(\Lambda ), \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q(\Lambda )=Q^0(\Lambda )+Q^1(\Lambda ) \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q^0(\Lambda )=\sum \{|\alpha _{j}||\beta _k|\,:\,\alpha _{j}\text { and }\beta _k\text { approach}\},\qquad Q^1(\Lambda )=|\alpha ||\Delta \sigma |, \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma =\sigma _{1}-\sigma _{2}$$\end{document} , and O(1) depends continuously on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} but independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , \beta , \Delta \sigma )$$\end{document} .

Fig. 5. Reflection at the boundary

Case 2

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b$$\end{document} covers the part of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }$$\end{document} but none of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x, \vartheta }$$\end{document} . We take three diamonds at the same time, as shown in Fig. 5. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{b,h}-1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{b,h}}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{b,h}+1}$$\end{document} denote the diamonds centering in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{b,h}-1})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{b,h}})$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{b,h}+1})$$\end{document} , respectively, and denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b=\Delta _{h,n_{b,h}-1}\cup \Delta _{h,n_{b,h}}\cup \Delta _{h,n_{b,h}+1}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} be the weak waves issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n_{b,h-1}-1})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n_{b,h-1}-2})$$\end{document} respectively, and entering \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b$$\end{document} . We divide \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =(\alpha _{1},\alpha _{2})$$\end{document} into parts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{l}=(\alpha _{l,1},0)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{r}=(\alpha _{r,1},\alpha _{r,2})$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{l}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{r}$$\end{document} entering \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{b,h}-1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{b,h}}$$\end{document} , respectively. Moreover, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(\nu _{1},\nu _{2})$$\end{document} , and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} be the outgoing wave issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{b,h}-1})$$\end{document} .

For simplicity of notation, we denote that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sigma _{\alpha }=\sigma (x_{h-1},y_{n_{b,h-1}-1}), & \sigma _{b}(h-1)=\sigma (x_{h-1},b_{\Delta x,\vartheta }(x_{h-1})),\\&\sigma _{b}(h)=\sigma (x_{h},b_{\Delta x,\vartheta }(x_{h})), & \sigma _{\nu }=\sigma (x_{h-1},y_{n_{b,h-1}-2}),\\&\sigma _{0}=\sigma (x_{h-1},r_{h-1,n_b-2}), & \Delta \sigma _{\alpha }=\sigma _{b}(h-1)-\sigma _{\alpha },\\&\Delta {\bar{\sigma }}_{\alpha }=\sigma _{b}(h)-\sigma _{\alpha }, & \Delta \sigma _{b_{h}}=\sigma _{b}(h)-\sigma _{b}(h-1),\\&\Delta \sigma _{\nu }= \sigma _{\alpha }-\sigma _{\nu }, & \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{1}=U_{\Delta x,\vartheta }(x_{h-1}-,r_{h-1,n_b-2}+)$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}=\Phi (\alpha _{l,1},0;{\tilde{U}}(\sigma _{\alpha };\sigma _{0},U_{1}))$$\end{document} .

To gain the estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} , we need to deal with the equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\frac{1}{2}|\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{b}(h);\sigma _{\alpha },U_{l}))|^2 +\frac{\gamma }{\gamma -1}(p^{b}_{\Delta x,h+1})^{\frac{\gamma -1}{\gamma }}\\&\quad = \frac{1}{2}|{\tilde{U}}(\sigma _{b}(h-1);\sigma _{\alpha },\Phi (\alpha _{r,1},\alpha _{r,2};U_{l}))|^2 +\frac{\gamma }{\gamma -1}(p^{b}_{\Delta x,h})^{\frac{\gamma -1}{\gamma }}, \end{aligned} \end{aligned}$$\end{document}

and then we obtain the following lemma:

Lemma 4.3

Equation (4.5) has a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{1}=\beta _{1}(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,\omega _{h+1};U_{l})\in \text {C}^2$$\end{document} in a neighborhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,\omega _{h+1},U_{l})=(0,0,0,0,0,G(s_{0}))$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{h+1}=p^{b}_{\Delta x,h+1}-p^{b}_{\Delta x,h}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \delta _{1}&=\alpha _{r,1}+\alpha _{l,1}+\nu _{1} +K_{r,1}\alpha _{r,2} +K_{\sigma ,1}\Delta \sigma _{b_{h}}+K_{b,1}\omega _{h+1}+O(1)Q(\Lambda _b),\\ \delta _{2}&=\nu _{2}+K_{r,2}\alpha _{r,2} +K_{\sigma ,2}\Delta \sigma _{b_{h}}+K_{b,2}\omega _{h+1}+O(1)Q(\Lambda _b), \end{aligned} \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q(\Lambda _b)=Q^0((\alpha _{1},0),\nu )+|\alpha _{1}||\Delta \sigma _\nu |+|\alpha _{r,1}||\Delta \sigma _\alpha |, \end{aligned}$$\end{document}

where O(1) depends continuously on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} . Moreover, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{r,1}=\alpha _{r,2}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{b}_{\Delta x,h+1}=p_{0}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{l}=G(s_{0})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }K_{r,1}=-\dfrac{\cos ^2(\theta _{0}+\theta _{m}^0)}{\cos ^2(\theta _{0}-\theta _{m}^0)},\ \lim _{M_{\infty }\rightarrow \infty }|K_{b,i}|<\infty ,\nonumber \\&\quad \lim _{M_{\infty }\rightarrow \infty }K_{r,2}=0,\qquad \lim _{M_{\infty }\rightarrow \infty }K_{\sigma ,i}=0, \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2$$\end{document} .

Proof

A direct computation leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left. \dfrac{1}{2} \frac{\partial (|\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{b}(h);\sigma _{\alpha },U_{l}))|^2)}{\partial \beta _{1}}\right| _{\{\delta _{1}=\Delta {\bar{\sigma }}_\alpha =0, U_{l}=G(s_{0})\}}=r_{1}(G(s_{0}))\cdot G(s_{0}). \end{aligned}$$\end{document}

Lemma 2.7, together with the implicit function theorem, implies that there is a unique \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {C}^2-$$\end{document} solution

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{1}=\beta _{1}(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,\omega _{h+1};U_{l}) \end{aligned}$$\end{document}

in a neighborhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,\omega _{h+1},U_{l})=(0,0,0,0,0,G(s_{0}))$$\end{document} .

Using (4.1)–(4.2), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{1}&=\beta _{1}(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta \sigma _\alpha ,\omega _{h+1};U_{l}) +{\bar{K}}_{\sigma ,1}(\Delta {\bar{\sigma }}_\alpha -\Delta \sigma _\alpha )\\&=\beta _{1}(\alpha _{r,1},0,\Delta \sigma _\alpha ,\Delta \sigma _\alpha ,0;U_{l}) +{\bar{K}}_{\sigma ,1}\Delta \sigma _{b_{h}}+{\bar{K}}_{r,1}\alpha _{r,2}+{\bar{K}}_{b,1}\omega _{h+1}\\&=\alpha _{r,1} +{\bar{K}}_{\sigma ,1}\Delta \sigma _{b_{h}}+{\bar{K}}_{r,1}\alpha _{r,2}+{\bar{K}}_{b,1}\omega _{h+1}+O(1)|\alpha _{r,1}||\Delta \sigma _\alpha |. \end{aligned}$$\end{document}

Taking the derivative with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma _{b_{h}}$$\end{document} in (4.5) at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,\omega _{h+1},U_{l})=(0,0,0,0,0,G(s_{0}))$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G(s_{0})\cdot r_{1}(G(s_{0}))\frac{\partial \beta _{1}}{\partial \Delta \sigma _{b_{h}}} +G(s_{0})\cdot \frac{\partial {\tilde{U}}(\sigma _\alpha +\Delta {\bar{\sigma }}_\alpha +\Delta \sigma _{b_{h}};\sigma _\alpha ,G(s_{0}))}{\partial \Delta \sigma _{b_{h}}}=0, \end{aligned}$$\end{document}

which yields

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty } \left. \frac{\partial \beta _{1}}{\partial \Delta \sigma _{b_{h}}}\right| _{\{\alpha _{r,1}=\alpha _{r,2}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0,\,p^{b}_{\Delta x,h+1}=p_{0},\,U_{l}=G(s_{0})\}}=0. \end{aligned}$$\end{document}

Similarly, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }\left. \frac{\partial \beta _{1}}{\partial \omega _{h+1}}\right| _{\{\alpha _{r,1}=\alpha _{r,2}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0,\,p^{b}_{\Delta x,h+1}=p_{0},\,U_{l}=G(s_{0})\}}\\ &\qquad =\lim _{M_{\infty }\rightarrow \infty }\dfrac{-p_{0}^{-\frac{1}{\gamma }}}{r_{1}(G(s_{0}))\cdot G(s_{0})}>-\infty ,\\&\lim _{M_{\infty }\rightarrow \infty }\left. \frac{\partial \beta _{1}}{\partial \alpha _{r,2}}\right| _{\{\alpha _{r,1}=\alpha _{r,2}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0,\,p^{b}_{\Delta x,h+1}=p_{0},\,U_{l}=G(s_{0})\}}\\ &\qquad =-\dfrac{\cos ^2(\theta _{0}+\theta _{m}^0)}{\cos ^2(\theta _{0}-\theta _{m}^0)}. \end{aligned} \end{aligned}$$\end{document}

By the construction of the approximate solution, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{\tilde{U}}(\sigma _{b}(h);\sigma _\nu ,\Phi (\delta _{1},\delta _{2};U_m))\\&\quad =\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{b}(h);\sigma _{\alpha },\Phi (\alpha _{l,1},0;{\tilde{U}}(\sigma _\alpha ;\sigma _\nu ,\Phi (\nu _{1},\nu _{2};U_m))))) \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_m=U_{\Delta x,\vartheta }(x_{h},y_{n_{b,h-1}-2}-)$$\end{document} . Then, a similar argument as to that in Case 1 gives (4.6)–(4.8). This completes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.4

In Case 2, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_{h}:=b_{\Delta x,\vartheta }'(x_{h}-)$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_{+}$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b'_{h+1}-b'_{h}=K_{c,2}\alpha _{r,2}+K_{c,\sigma }\Delta \sigma _{b_{h}}+O(1)\omega _{h+1}+O(1)|\alpha _{r,1}||\Delta \sigma _{\alpha }| \end{aligned}$$\end{document}

with O(1) depending continuously on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{0}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }K_{c,\sigma }|_{\{\alpha _{r,2}=\alpha _{r,1}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0, p^{b}_{\Delta x,h+1}=p_{0},U_{l}=G(s_{0})\}}=-1,\\&\lim _{M_{\infty }\rightarrow \infty }K_{c,2}|_{\{\alpha _{r,2}=\alpha _{r,1}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0, p^{b}_{\Delta x,h+1}=p_{0},U_{l}=G(s_{0})\}}\\&\quad \,=-\frac{4}{\gamma +1}\,\frac{\cos ^2\theta ^0_m\cos ^2(\theta ^0_m+\theta _{0})}{\cos ^2\theta _{0}}. \end{aligned}$$\end{document}

Proof

From (3.2), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b'_{h+1}-b'_{h}&=\frac{\Phi ^{(2)}(\beta _{1}(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,\omega _{h+1};U_{l}),0; {\tilde{U}}(\sigma _{b_{h}};\sigma _{\alpha },U_{l}))}{\Phi ^{(1)}(\beta _{1}(\alpha _{r,1},\alpha _{r,2},\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,\omega _{h+1};U_{l}),0;{\tilde{U}}(\sigma _{b_{h}};\sigma _{\alpha },U_{l}))}\\&\quad -\frac{{\tilde{U}}^{(2)}(\sigma _{b_{h-1}};\sigma _{\alpha },\Phi (\alpha _{r,1},\alpha _{r,2};U_{l}))}{{\tilde{U}}^{(1)}(\sigma _{b_{h-1}};\sigma _{\alpha },\Phi (\alpha _{r,1},\alpha _{r,2};U_{l}))}. \end{aligned}$$\end{document}

By (4.1)–(4.2), we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&b'_{h+1}-b'_{h}\\ &\quad =\frac{\Phi ^{(2)}(\beta _{1}(\alpha _{r,1},0,\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,0;U_{l}),0;{\tilde{U}}(\sigma _{b_{h}};\sigma _{\alpha },U_{l}))}{\Phi ^{(1)}(\beta _{1}(\alpha _{r,1},0,\Delta \sigma _\alpha ,\Delta {\bar{\sigma }}_\alpha ,0;U_{l}),0;{\tilde{U}}(\sigma _{b_{h}};\sigma _{\alpha },U_{l}))}\\ &\qquad \, -\frac{{\tilde{U}}^{(2)}(\sigma _{b_{h-1}};\sigma _{\alpha },\Phi (\alpha _{r,1},0;U_{l}))}{{\tilde{U}}^{(1)}(\sigma _{b_{h-1}};\sigma _{\alpha },\Phi (\alpha _{r,1},0;U_{l}))}\\ &\quad \quad \, +K_{c,2}\alpha _{r,2}+O(1)\omega _{h+1}\\ &\quad =\frac{\Phi ^{(2)}(\beta _{1}(\alpha _{r,1},0,\Delta \sigma _\alpha ,\Delta \sigma _\alpha ,0;U_{l}),0;{\tilde{U}}(\sigma _{b_{h-1}};\sigma _{\alpha },U_{l}))}{\Phi ^{(1)}(\beta _{1}(\alpha _{r,1},0,\Delta \sigma _\alpha ,\Delta \sigma _\alpha ,0;U_{l}),0;{\tilde{U}}(\sigma _{b_{h-1}};\sigma _{\alpha },U_{l}))}\\ &\qquad \, -\frac{{\tilde{U}}^{(2)}(\sigma _{b_{h-1}};\sigma _{\alpha },\Phi (\alpha _{r,1},0;U_{l}))}{{\tilde{U}}^{(1)}(\sigma _{b_{h-1}};\sigma _{\alpha },\Phi (\alpha _{r,1},0;U_{l}))}\\ &\quad \quad \, +K_{c,\sigma }\Delta \sigma _{b_{h}}+K_{c,2}\alpha _{r,2}+O(1)\omega _{h+1}\\ &\quad =K_{c,\sigma }\Delta \sigma _{b_{h}}+K_{c,2}\alpha _{r,2}+O(1)\omega _{h+1}+O(1)|\alpha _{r,1}||\Delta \sigma _{\alpha }|. \end{aligned} \end{aligned}$$\end{document}

By similar calculation in Lemma 4.3, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }K_{c,\sigma }|_{\{\alpha _{r,2}=\alpha _{r,1}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0, \,p^{b}_{\Delta x,h+1}=p_{0},\,U_{l}=G(s_{0})\}}=-1,\\&\lim _{M_{\infty }\rightarrow \infty }K_{c,2}|_{\{\alpha _{r,2}=\alpha _{r,1}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0, \,p^{b}_{\Delta x,h+1}=p_{0},\,U_{l}=G(s_{0})\}}\\&\quad =-\frac{4}{\gamma +1}\,\frac{\cos ^2\theta ^0_m\cos ^2(\theta ^0_m+\theta _{0})}{\cos ^2\theta _{0}}. \end{aligned}$$\end{document}

Then the proof is complete. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.5

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} sufficiently small,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |b'_{h}-\sigma _{b}(h-1)| \geqq 6|\Delta \sigma _{b_h}|, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b}(h)=\frac{b_{\Delta x,\vartheta }(x_{h})}{x_{h}}$$\end{document} .

Proof

Using the notation as in Case 2, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b'_{h}=\dfrac{b_{\Delta x,\vartheta }(x_{h})-b_{\Delta x,\vartheta }(x_{h-1})}{\Delta x}. \end{aligned}$$\end{document}

Then a direct computation leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |b'_h-\sigma _{b}(h-1)|&=\left| \dfrac{b_{\Delta x,\vartheta }(x_{h})-b_{\Delta x,\vartheta }(x_{h-1})}{\Delta x}-\sigma _{b}(h-1)\right| \\&=\left| \dfrac{\sigma _{b}(h)x_{h}-\sigma _{b}(h-1)x_{h-1}}{\Delta x}-\sigma _{b}(h-1)\right| \\&=\left| \dfrac{x_{h}}{\Delta x}\right| |\sigma _{b}(h)-\sigma _{b}(h-1)|\\&\geqq 6\left| \sigma _{b}(h)-\sigma _{b}(h-1)\right| \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} small enough. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{b}(h)=|\sigma _{b}(h-1)-b'_{h}|$$\end{document} that measures the angle between boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta ,h}$$\end{document} and the ray issuing from the origin and passing through \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},b_{\Delta x,\vartheta }(x_{h-1}))$$\end{document} . Then we have the following estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{b}(h)$$\end{document} :

Lemma 4.6

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} sufficiently large and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} sufficiently small,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _{b}(h)-\theta _{b}(h+1)\geqq |\Delta \sigma |-|K_{c,2}||\alpha _{r,2}|-C|\omega _{h+1}|-C|\alpha _{r,1}||\Delta \sigma _{\alpha }|, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_{+}$$\end{document} , and constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} is independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} .

Proof

We consider the following two different cases:

1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b}(h-1)<b'_{h}$$\end{document} so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b}(h)>\sigma _{b}(h-1)$$\end{document} . 

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_{h+1}>\sigma _{b}(h)$$\end{document} , then it follows from Lemma 4.4 that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\theta _{b}(h)-\theta _{b}(h+1)\\&\quad = b'_{h}-\sigma _{b}(h-1)-\big (b'_{h+1}-\sigma _{b}(h)\big )\\&\quad = (1-K_{c,\sigma })\Delta \sigma _{b_{h}}-K_{c,2}\alpha _{r,2}+O(1)\omega _{h+1}+O(1)|\alpha _{r,1}||\Delta \sigma _{\alpha }|\\&\quad \geqq |\Delta \sigma _{b_{h}}|-|K_{c,2}||\alpha _{r,2}|-C|\omega _{h+1}|-C|\alpha _{r,1}||\Delta \sigma _{\alpha }|. \end{aligned}$$\end{document}

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_{h+1}<\sigma _{b}(h)$$\end{document} , then, from Lemma 4.4–4.5, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\theta _{b}(h)-\theta _{b}(h+1)\\&\quad = b'_{h}-\sigma _{b}(h-1)-\big (\sigma _{b}(h)-b'_{h+1}\big )\\&\quad = 2\big (b'_{h}-\sigma _{b}(h-1)\big )+b'_{h+1}\\&\qquad -\sigma _{b}(h)-\big (b'_{h}-\sigma _{b}(h-1)\big )\\&\quad \geqq (11+K_{c,\sigma })|\Delta \sigma _{b_{h}}|+K_{c,2}\alpha _{r,2}+O(1)\omega _{h+1}+O(1)|\alpha _{r,1}||\Delta \sigma _{\alpha }|\\&\quad \geqq |\Delta \sigma _{b_{h}}|-|K_{c,2}||\alpha _{r,2}|-C|\omega _{h+1}|-C|\alpha _{r,1}||\Delta \sigma _{\alpha }|. \end{aligned}$$\end{document}

2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b}(h-1)>b'_{h}$$\end{document} so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{b}(h)<\sigma _{b}(h-1)$$\end{document} . 

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_{h+1}>\sigma _{b}(h)$$\end{document} , then it follows from Lemma 4.4–4.5 that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\theta _{b}(h)-\theta _{b}(h+1)\\&\quad =\sigma _{b}(h-1)-b'_{h}-\big (b'_{h+1}-\sigma _{b}(h)\big )\\&\quad = 2(\sigma _{b}(h-1)-b'_{h})+\sigma _{b}(h)-b'_{h+1}-(\sigma _{b}(h-1)-b'_{h})\\&\quad \geqq (11+K_{c,\sigma })|\Delta \sigma _{b_{h}}|-K_{c,2}\alpha _{r,2}-O(1)\omega _{h+1}-O(1)|\alpha _{r,1}||\Delta \sigma _{\alpha }|\\&\quad \geqq |\Delta \sigma _{b_{h}}|-|K_{c,2}||\alpha _{r,2}|-C|\omega _{h+1}|-C|\alpha _{r,1}||\Delta \sigma _{\alpha }|. \end{aligned}$$\end{document}

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_{h+1}<\sigma _{b}(h)$$\end{document} , then, from Lemma 4.4, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\theta _{b}(h)-\theta _{b}(h+1)\\&\quad =\sigma _{b}(h-1)-b'_{h}-\big (\sigma _{b}(h)-b'_{h+1}\big )\\&\quad =(-1+K_{c,\sigma })\Delta \sigma _{b_{h}}+K_{c,2}\alpha _{r,2}+O(1)\omega _{h+1}+O(1)|\alpha _{r,1}||\Delta \sigma _{\alpha }|\\&\quad \geqq |\Delta \sigma _{b_{h}}|-|K_{c,2}||\alpha _{r,2}|-C|\omega _{h+1}| -C|\alpha _{r,1}||\Delta \sigma _{\alpha }|. \end{aligned}$$\end{document}

Note that we have used the fact that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }K_{c,\sigma }|_{\{\alpha _{r,2}=\beta _{1}=\Delta \sigma _\alpha =\Delta {\bar{\sigma }}_\alpha =\omega _{h+1}=0, p^{b}_{\Delta x,h+1}=p_{0},U_{l}=G(s_{0})\}}=-1 \end{aligned}$$\end{document}

in the above estimates. This completes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Fig. 6. Near the strong shock wave

Case 3

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{s}$$\end{document} covers the part of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x,\vartheta }$$\end{document} but none of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x, \vartheta }$$\end{document} . We take three diamonds at the same time, as shown in Fig. 6. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{\chi ,h}-1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{\chi ,h}}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{\chi ,h}+1}$$\end{document} be the diamonds centering in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{\chi ,h}-1})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{\chi ,h}})$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{\chi ,h}+1})$$\end{document} , respectively. Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{s}=\Delta _{h,n_{\chi ,h}-1}\cup \Delta _{h,n_{\chi ,h}}\cup \Delta _{h,n_{\chi ,h}+1}$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} be the weak waves issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n_{\chi ,h-1}+1})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n_{\chi ,h-1}+2})$$\end{document} respectively and entering \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{s}$$\end{document} . We divide \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} into parts \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{l}=(\alpha _{l,1},0)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{r}=(\alpha _{r,1},\alpha _{r,2})$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{l}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{r}$$\end{document} enter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{\chi ,h}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{h,n_{\chi ,h}+1}$$\end{document} , respectively. Moreover, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =(\nu _{1},0)$$\end{document} , and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} be the outgoing wave issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_{n_{\chi ,h}+1})$$\end{document} .

Then, for simplicity of notation, we denote that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sigma _{\alpha }=\sigma (x_{h-1},y_{n_{\chi ,h-1}+1}), & \sigma _{\chi }(h-1)=\sigma (x_{h-1},\chi _{\Delta x,\vartheta }(x_{h-1})),\\&\sigma _{\chi }(h)=\sigma (x_{h},\chi _{\Delta x,\vartheta }(x_{h})), & \sigma _{\nu }=\sigma (x_{h-1},y_{n_{\chi ,h-1}+2}),\\&\Delta \sigma _{\alpha }=\sigma _{\alpha }-\sigma _{\chi }(h-1), & \Delta {\bar{\sigma }}_{\alpha }=\sigma _{\alpha }-\sigma _{\chi }(h),\\&\Delta \sigma _{\chi _{h}}=\sigma _{\chi }(h)-\sigma _{\chi }(h-1), & \Delta \sigma _{\nu }= \sigma _{\nu }-\sigma _{\alpha }.&\end{aligned}$$\end{document}

To gain the estimates of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s_{h+1},\delta )$$\end{document} , we need to deal with the equation:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\tilde{U}}(\sigma _\alpha ;\sigma _{\chi }(h),\Phi (0,\beta _{2};G(s_{h+1};U_{\infty }))) =\Phi (\alpha _{l,1},0;{\tilde{U}}(\sigma _\alpha ;\sigma _{\chi }(h-1),G(s_{h};U_{\infty }))), \end{aligned}$$\end{document}

to obtain the following lemma:

Lemma 4.7

Equation (4.10) has a unique solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s_{h+1},\beta _{2})$$\end{document} in a neighborhood of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\alpha _{l,1},\alpha _{r},\nu ,\Delta \sigma _{\alpha },\Delta \sigma _{\chi _{h}},s_{h})=(0,0,0,0,0,s_{0}) \end{aligned}$$\end{document}

such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\delta _{1}=\alpha _{r,1}+\nu _{1}+\mu _{w,1}\Delta \sigma _{\chi _{h}}+K_{w,1}\alpha _{l,1}+O(1)Q(\Lambda _{s}),\\&\delta _{2}=\alpha _{r,2}+\mu _{w,2}\Delta \sigma _{\chi _{h}}+K_{w,2}\alpha _{l,1}+O(1)Q(\Lambda _{s}),\\&s_{h+1}= s_{h}+K_{s}\alpha _{l,1}+\mu _{s}\Delta \sigma _{\chi _{h}}, \end{aligned} \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q(\Lambda _{s})=|\nu _{1}||\Delta \sigma _{\nu }|+Q^0(\alpha _{r},\nu ), \end{aligned}$$\end{document}

where O(1) depends continuously on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} . In addition, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{l}=0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma _{\alpha }=\Delta \sigma _{\chi _{h}}=0$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{h}=s_{0}$$\end{document} , denoting the derivative of G by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s}$$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }K_{w,1}=0, & K_{w,2}=\dfrac{\det (r_1(G(s_0)),\,G_{s}(s_0))}{\det (r_2(G(s_0)),\,G_{s}(s_0))},\\&K_s=\dfrac{\det (r_2(G(s_0)),r_1(G(s_0)))}{\det (r_2(G(s_0)),G_{s}(s_0))}, & \qquad \qquad \qquad \lim _{M_{\infty }\rightarrow \infty }\mu _s\in (-1,0),\\&\lim _{M_{\infty }\rightarrow \infty }\mu _{w,1}=0, & \mu _{w,2}=\dfrac{\det (\partial {\tilde{U}}/\partial (\Delta \sigma _{\chi _h}),\,G_{s}(s_0))}{\det (r_2(G(s_0)),\,G_{s}(s_0))}. \end{aligned} \end{aligned}$$\end{document}

Proof

From Lemma 2.10 and the implicit function theorem, (4.10) has a unique \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {C}^2$$\end{document} –solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s_{h+1},\beta _{2})$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&s_{h+1}=s_{h+1}(\alpha _{l,1},\Delta \sigma _{\alpha },\Delta {\bar{\sigma }}_{\alpha },\Delta \sigma _{\chi _{h}},s_{h}),\\&\beta _{2}=\beta _{2}(\alpha _{l,1},\Delta \sigma _{\alpha },\Delta {\bar{\sigma }}_{\alpha },\Delta \sigma _{\chi _{h}},s_{h}). \end{aligned}$$\end{document}

A direct computation leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{2}&=\beta _{2}(\alpha _{l,1},\Delta \sigma _{\alpha },\Delta {\bar{\sigma }}_{\alpha },\Delta \sigma _{\chi _{h}},s_{h})\\&=\mu _{w,2}\Delta \sigma _{\chi _{h}}+K_{w,2}\alpha _{l,1} +\beta _{2}(0,\Delta \sigma _{\alpha },\Delta \sigma _{\alpha },0,s_{h})\\&=\mu _{w,2}\Delta \sigma _{\chi _{h}}+K_{w,2}\alpha _{l,1}. \end{aligned}$$\end{document}

Similarly, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s_{h+1}=s_{h+1}(\alpha _{l,1},\Delta \sigma _{\alpha },\Delta {\bar{\sigma }}_{\alpha },\Delta \sigma _{\chi _{h}},s_{h}) =\mu _s\Delta \sigma _{\chi _{h}}+K_{s}\alpha _{l,1}+s_{h}. \end{aligned}$$\end{document}

Next, we compute the coefficients: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_s$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{w,2}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{w,2}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _s$$\end{document} . Differentiating equation (4.10) with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{l,1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma _{\chi _{h}}$$\end{document} , and then letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{l,1}=\Delta \sigma _{\alpha }=\Delta \sigma _{\chi _{h}}=0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{h}=s_{0}$$\end{document} , we can obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&r_{2}(G(s_{0}))K_{w,2}+G_s(s_{0})K_s=r_{1}(G(s_{0})),\\&r_{2}(G(s_{0}))\mu _{w,2}+G_s(s_{0})\mu _s =\dfrac{\partial {\tilde{U}}}{\partial (\Delta \sigma _{\chi _{h}})}(\sigma _{\chi }(h);\sigma _{\chi }(h),G(s_{0})). \end{aligned}$$\end{document}

Then Cramer’s rule gives the result. Moreover, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{0}<0<\theta ^0_{ma}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{0}\pm \theta ^0_{ma}\in (-\frac{\pi }{2},\frac{\pi }{2})$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }\mu _s=\frac{\cos \theta _{0}\sin \theta _{m}^0}{\sin (\theta _{0}-\theta _{m}^0)}\in (-1,0). \end{aligned}$$\end{document}

By the construction of the approximate solution, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{\tilde{U}}(\sigma _{\nu };\sigma _{\alpha },\Phi (\delta _{1},\delta _{2};{\tilde{U}}(\sigma _\alpha ;\sigma _{\chi }(h),U_m)))\\&\quad =\Phi (\nu _{1},0; {\tilde{U}}(\sigma _{\nu };\sigma _{\alpha }, \Phi (\alpha _{r,1},\alpha _{r,2}; {\tilde{U}}(\sigma _\alpha ;\sigma _{\chi }(h), \Phi (0,\beta _{2}(\alpha _{l,1},\Delta \sigma _{\alpha },\Delta {\bar{\sigma }}_{\alpha },\Delta \sigma _{\chi _{h}},s_{h});U_m))))), \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} U_m=G(s_{h+1};U_{\infty }). \end{aligned}$$\end{document}

Then, by similar arguments as to those in Case 1, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta _{1}&=\alpha _{r,1}+\nu _{1}+\mu _{w,1}\Delta \sigma _{\chi _{h}}+K_{w,1}\alpha _{l,1}+O(1)|\nu _{1}||\Delta \sigma _{\nu }|+O(1)Q^0(\alpha _{r},\nu ),\\ \delta _{2}&=\alpha _{r,2}+\mu _{w,2}\Delta \sigma _{\chi _{h}}+K_{w,2}\alpha _{l,1}+O(1)|\nu _{1}||\Delta \sigma _{\nu }|+O(1)Q^0(\alpha _{r},\nu ). \end{aligned}$$\end{document}

This completes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.8

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} sufficiently small,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |s_h-\sigma _{\chi }(h-1)|\geqq 6|\Delta \sigma _{\chi _{h}}|. \end{aligned}$$\end{document}

Proof

Using the notation as in Case 3, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma _{\chi }(h)=\frac{\chi _{\Delta x,\vartheta }(x_{h})}{x_{h}},\qquad s_{h}=\dfrac{\chi _{\Delta x,\vartheta }(x_{h})-\chi _{\Delta x,\vartheta }(x_{h-1})}{\Delta x}. \end{aligned}$$\end{document}

Then a direct computation leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |s_h-\sigma _{\chi }(h-1)|&=\left| \dfrac{\chi _{\Delta x,\vartheta }(x_{h})-\chi _{\Delta x,\vartheta }(x_{h-1})}{\Delta x}-\sigma _{\chi }(h-1)\right| \\&=\left| \dfrac{\sigma _{\chi }(h)x_{h}-\sigma _{\chi }(h-1)x_{h-1}}{\Delta x}-\sigma _{\chi }(h-1)\right| \\&=\left| \dfrac{x_{h}}{\Delta x}\right| |\sigma _{\chi }(h)-\sigma _{\chi }(h-1)|\\&\geqq 6\left| \sigma _{\chi }(h)-\sigma _{\chi }(h-1)\right| , \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} small enough. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Denote \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{\chi }(h)=|\sigma _{\chi }(h-1)-s_{h}|$$\end{document} that measures the angle between the leading shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x,\vartheta ,h}$$\end{document} and the ray issuing from the origin and passing through \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},\chi _{\Delta x,\vartheta }(x_{h-1}))$$\end{document} . Then we have the following estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{\chi }(h)$$\end{document} :

Lemma 4.9

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} sufficiently large and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} sufficiently small,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _{\chi }(h)-\theta _{\chi }(h+1)\geqq |\Delta \sigma _{\chi _{h}}|-|K_s||\alpha _{l,1}|, \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\geqq 0$$\end{document} .

Proof

We consider the following two different cases:

1. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\chi }(h-1)<s_{h}$$\end{document} so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\chi }(h)>\sigma _{\chi }(h-1)$$\end{document} . 

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{h+1}>\sigma _{\chi }(h)$$\end{document} , then it follows from Lemma 4.7 that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _{\chi }(h)-\theta _{\chi }(h+1)&= s_{h}-\sigma _{\chi }(h-1)-\big (s_{h+1}-\sigma _{\chi }(h)\big )\\&= (1-\mu _s)\Delta \sigma _{\chi _{h}}-K_s\alpha _{l,1}\\&\geqq |\Delta \sigma _{\chi _{h}}|-|K_s||\alpha _{l,1}|. \end{aligned}$$\end{document}

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{h+1}<\sigma _{\chi }(h)$$\end{document} , then, from Lemmas 4.7–4.8, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _{\chi }(h)-\theta _{\chi }(h+1)&=s_{h}-\sigma _{\chi }(h-1)-\big (\sigma _{\chi }(h)-s_{h+1}\big )\\&= 2(s_{h}-\sigma _{\chi }(h-1))+s_{h+1}-\sigma _{\chi }(h)-\big (s_{h}-\sigma _{\chi }(h-1)\big )\\&\geqq (11+\mu _s)|\Delta \sigma _{\chi _{h}}|+K_s\alpha _{l,1}\\&\geqq |\Delta \sigma _{\chi _{h}}|-|K_s||\alpha _{l,1}|. \end{aligned}$$\end{document}

2. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\chi }(h-1)>s_{h}$$\end{document} so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\chi }(h)<\sigma _{\chi }(h-1)$$\end{document} . 

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{h+1}>\sigma _{\chi }(h)$$\end{document} , then it follows from Lemmas 4.7–4.8 that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\theta _{\chi }(h)-\theta _{\chi }(h+1)\\&\quad =\sigma _{\chi }(h-1)-s_{h}-\big (s_{h+1}-\sigma _{\chi }(h)\big )\\&\quad =2(\sigma _{\chi }(h-1)-s_{h})+\sigma _{\chi }(h)-s_{h+1}-(\sigma _{\chi }(h-1)-s_{h})\\&\quad \geqq (11+\mu _s)|\Delta \sigma _{\chi _{h}}|-K_s\alpha _{l,1}\\&\quad \geqq |\Delta \sigma _{\chi _{h}}|-|K_s||\alpha _{l,1}|. \end{aligned}$$\end{document}

• If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{h+1}<\sigma _{\chi }(h)$$\end{document} , then, from Lemma 4.7, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \theta _{\chi }(h)-\theta _{\chi }(h+1)&=\sigma _{\chi }(h-1)-s_{h}-\big (\sigma _{\chi }(h)-s_{h+1}\big )\\&=(-1+\mu _s)\Delta \sigma _{\chi _{h}}+K_{s}\alpha _{l,1}\\&\geqq |\Delta \sigma _{\chi _{h}}|-|K_s||\alpha _{l,1}|. \end{aligned}$$\end{document}

Note that we have used the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _s\in (-1,0)$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }\rightarrow \infty $$\end{document} in above estimates. This completes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Glimm-Type Functional and Compactness of the Approximate Solutions

For each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\subset \cup _{k=1}^{h+1}\Omega _{\Delta x,\vartheta ,k}$$\end{document} , there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{I}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\leqq k_{I}\leqq h+1$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\cap \Gamma _{\Delta x,\vartheta , k_{I}}\ne \emptyset $$\end{document} . Next, as in [38, 51], we assign each mesh curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\subset \bigcup _{k=1}^{h+1}\Omega _{\Delta x,\vartheta ,k}$$\end{document} with a Glimm-type functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_s(I)$$\end{document} ; see also [10, 48].

Definition 5.1

(Weighted total variation).  Define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&L_{0}^{(i)}(I)=\sum \{|\alpha _{i}|:\alpha _{i} \text { is the weak }i\text {-wave crossing }I\} \qquad \text{ for } i=1,\,2,\\&L_{1}(I)=\sum \{|\omega _{k}|: k>k_{I}\},\\&L_s(I)=\theta _\chi (I) \quad \text { for } \theta _{\chi }(I)=\theta _\chi (h)\text { in Lemma }4.9 \text { when }S_{\Delta x,\vartheta }\text { crossing }I,\\&L_b(I)=\theta _b(I) \quad \,\text { for }\theta _{b}(I)=\theta _b(h)\text { in Lemma }4.6\text { when }\Gamma _{\Delta x,\vartheta } \text { crossing }I. \end{aligned}$$\end{document}

Then the weighted total variation is defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(J)=L_{0}^{(1)}(I)+K_{2}L_{0}^{(2)}(I)+K_{1}L_{1}(I)+K_3L_s(I)+K_4L_b(I), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{l}$$\end{document} are positive constants for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l=1,2,3,4$$\end{document} .

Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma ^*=b_{0}+C_{1}\sum _{h=1}^{\infty }|\omega _{h}|,\qquad \sigma _*=s_{0}-\varpi , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{0}$$\end{document} is the velocity of the leading shock of the background solution, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varpi $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1}$$\end{document} are constants to be determined; see also [10, 18, 48]. Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varpi $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{h\geqq 1}|\omega _{h}|$$\end{document} are chosen so small that the largeness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} implies the smallness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{0}-s_{0}$$\end{document} , which leads to the smallness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*-\sigma _*$$\end{document} . We now define the total interaction potential.

Definition 5.2

(Total interaction potential).  Define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&Q_{0}(I) =\sum \{|\alpha ||\beta |\,:\,\alpha \text { and }\beta \text { are weak waves crossing }I\text { and approach}\},\\&Q_{1}(I) =\sum \{|\alpha ||\sigma _\alpha -\sigma _*|\,:\,\alpha \text { is a weak }1-\text {wave crossing }I\},\\&Q_{2}(I)=\sum \{|\alpha ||\sigma ^*-\sigma _\alpha |\,:\,\alpha \text { is a weak }2-\text {wave crossing }I\}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _\alpha $$\end{document} is the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} -coordinate of the grid point where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} issues. Then the total interaction potential is defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q(I)=Q_{0}(I)+2Q_{1}(I)+2Q_{2}(I). \end{aligned}$$\end{document}

Now, we are able to define the Glimm-type functional.

Definition 5.3

(Glimm-type functional).  Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(I)=L(I)+KQ(I), \end{aligned}$$\end{document}

where K is a large real number to be chosen later.

Let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} E_{\Delta x,\vartheta }(\Lambda )=\left\{ \begin{aligned}&Q(\Lambda ) & (\text {defined in Case }1),&\\&\xi \big (|\alpha _{r,2}|+|\omega _{h+1}|+|\Delta \sigma _{b_{h}}|+Q(\Lambda _b)\big ) & (\text {defined in Case }2),&\\&\xi \big (|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|+Q(\Lambda _s)\big ) & (\text {defined in Case }3),&\end{aligned} \right. \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi >0$$\end{document} sufficiently small and to be chosen later.

In order to make the Glimm-type functional monotonically decreasing, we have to choose the weights carefully in the functional, based on the underlying features of the wave interactions governed by the system. Indeed, we have the following lemma (cf. [48]):

Lemma 5.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{r,1}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{w,2}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_s$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{w,2}$$\end{document} be given by Lemmas 4.3 and 4.7. Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty } \big (|K_{r,1}||K_{w,2}|+|K_{r,1}||K_s||\mu _{w,2}|\big )<1. \end{aligned}$$\end{document}

Proof

Lemmas 2.8–2.9 give

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }|K_{r,1}||K_{w,2}|=\left| \dfrac{\sin (\theta _{0}+\theta _{m}^0)}{\sin (\theta _{0}-\theta _{m}^0)} \right| ,\\&\lim _{M_{\infty }\rightarrow \infty }|K_{r,1}||K_s||\mu _{w,2}|\\&\quad =\dfrac{\cos ^2(\theta _{0}+\theta _{m}^0)}{\cos ^2(\theta _{0}-\theta _{m}^0)} \\&\qquad \times \lim _{M_{\infty }\rightarrow \infty } \left| \dfrac{\det \big (r_{1}(U),r_{2}(U)\big )}{\det \big (r_{2}(G(s_{0}),G_s(s_{0}))\big )} \right| \left| \dfrac{\det ((\partial {\tilde{U}})/(\partial \Delta \sigma _{\chi _{h}}),G_s(s_{0};U_{\infty }))}{\det (r_{2}(G(s_{0};U_{\infty })),G_s(s_{0};U_{\infty }))}\right| \\&\quad =\dfrac{\sin 2\theta _{m}^0\cos \theta _{0}|\sin \theta _{0}|}{\sin ^2(\theta _{0}-\theta _{m}^0)}. \end{aligned}$$\end{document}

Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{0}\in (-\frac{\pi }{2},0)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{0}\pm \theta _{m}^0\in (-\frac{\pi }{2},\frac{\pi }{2})$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{m}^0\in (0,\frac{\pi }{2})$$\end{document} . Then, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{0}+\theta _{m}^0<0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty } \big (|K_{r,1}||K_{w,2}|+|K_{r,1}||K_s||\mu _{w,2}|\big ) <\dfrac{2\sin \theta _{m}^0\cos \theta _{0}-\sin (\theta _{0}+\theta _{m}^0)}{\sin (\theta _{m}^0-\theta _{0})}=1; \end{aligned}$$\end{document}

when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{0}+\theta _{m}^0>0$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{M_{\infty }\rightarrow \infty }\big (|K_{r,1}||K_{w,2}|+|K_{r,1}||K_s||\mu _{w,2}|\big ) <\dfrac{2\cos \theta _{m}^0|\sin \theta _{0}|+\sin (\theta _{0}+\theta _{m}^0)}{\sin (\theta _{m}^0-\theta _{0})}=1. \end{aligned}$$\end{document}

This implies the expected result. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

At this stage, we are able to choose the coefficients in the Glimm-type functional (cf. [48]).

Lemma 5.2

There exist positive constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_3$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{M_{\infty }\rightarrow \infty }\big (K_{2}|K_{w,2}|+K_3|K_s|\big )<1, & \lim _{M_{\infty }\rightarrow \infty }\big (K_{2}|\mu _{w,2}|-K_3\big )<0,\\&\lim _{M_{\infty }\rightarrow \infty }\big (K_{2}-|K_{r,1}|\big )>0. & \end{aligned}$$\end{document}

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{r,1}^*=\lim _{M_{\infty }\rightarrow \infty }|K_{r,1}|$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{w,2}^*=\lim _{M_{\infty }\rightarrow \infty }|K_{w,2}|$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_s^*=\lim _{M_{\infty }\rightarrow \infty }|K_s|$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{w,2}^*=\lim _{M_{\infty }\rightarrow \infty }|\mu _{w,2}|$$\end{document} . Then, by Lemma 5.1,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K_{r,1}^*\big (K_{w,2}^*+K_s^*\mu _{w,2}^*\big )<1. \end{aligned}$$\end{document}

Hence, we choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{2}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K_{2}>K_{r,1}^*,\qquad K_{2}\big (K_{w,2}^*+K_s^*\mu _{w,2}^*\big )<1, \end{aligned}$$\end{document}

which implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K_{2}K_s^*\mu _{w,2}^*<1-K_{2}K_{w,2}^*. \end{aligned}$$\end{document}

Then we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_3$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K_3>K_{2}\mu _{w,2}^*,\qquad K_3K_s^*<1-K_{2}K_{w,2}^*, \end{aligned}$$\end{document}

and the proof is complete. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

With the coefficients chosen properly, we can derive a decay property for the Glimm-type functional.

Proposition 5.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} be sufficiently large, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^{*}-\sigma _{*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{h\geqq 1}|\omega _{h}|$$\end{document} be sufficiently small. Let I and J be a pair of space-like mesh curves with J being an immediate successor of I. The region bounded by the difference between I and J is denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} . Then there exist positive constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _\infty $$\end{document} , K, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{l}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l=1,2,3,4$$\end{document} , such that, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(I)<\varepsilon _\infty $$\end{document} , then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(I)\leqq F(J)-\frac{1}{4}E_{\Delta x, \vartheta }(\Lambda ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\Delta x, \vartheta }(\Lambda )$$\end{document} is given by (5.2).

Proof

When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} is large enough, according to Lemma 5.2, there are constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{2}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_3$$\end{document} so that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&K_{2}|K_{w,2}|+K_3|K_s|<1-\xi _{0},\quad K_{2}|\mu _{w,2}|-K_3<-\xi _{0},\quad \\&K_{2}-|K_{r}|-K_4|K_{c,2}|>\xi _{0} \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _{0}>0$$\end{document} .

Now, as in [38], we prove the result inductively; see also [10, 48]. We consider three special cases as in §4, depending on the location of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} . From now on, we use C to denote a universal constant depending only on the system, which may be different at each occurrence.

Case 1.  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} lies between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x, \vartheta }$$\end{document} . We consider the case as in Lemma 4.1. Notice that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&(L_{0}^{(1)}+K_{2}L_{0}^{(2)})(J)-(L_{0}^{(1)}+K_{2}L_{0}^{(2)})(I)\leqq CQ(\Lambda ),\\&L_b(J)-L_b(I)=0,\\&(K_{1}L_{1}+K_3L_s)(J)-(K_{1}L_{1}+K_3L_s)(I)=0. \end{aligned}$$\end{document}

Then we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(J)-L(I)\leqq CQ(\Lambda ). \end{aligned}$$\end{document}

For the terms contained in Q, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q_{0}(J)-Q_{0}(I)\leqq CL(I)Q(\Lambda )-Q^0(\Lambda ). \end{aligned}$$\end{document}

For Case 1.1:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (Q_{1}+Q_{2})(J)-(Q_{1}+Q_{2})(I)&{=} |\delta _{1}|(\sigma _{2}{-}\sigma _*){-}|\alpha _{1}|(\sigma _{2}-\sigma _*){-}|\beta _{1}|(\sigma _{1}-\sigma _*)\\&\quad +|\delta _{2}|(\sigma ^*-\sigma _{2})-|\alpha _{2}|(\sigma ^*-\sigma _{2})\\&\leqq C(\sigma ^*-\sigma _*)Q(\Lambda )-|\Delta \sigma ||\beta _{1}|. \end{aligned}$$\end{document}

For Case 1.2:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (Q_{1}+Q_{2})(J)-(Q_{1}+Q_{2})(I)&\leqq |\delta _{1}|(\sigma _{1}-\sigma _*)-|\beta _{1}|(\sigma _{1}-\sigma _*)\\&\quad +|\delta _{2}|(\sigma ^*-\sigma _{1})-|\alpha _{2}|(\sigma ^*-\sigma _{2})\\&\quad -|\beta _{2}|(\sigma ^*-\sigma _{1})\\&\leqq C(\sigma ^*-\sigma _*)Q(\Lambda )-|\Delta \sigma ||\alpha _{2}|, \end{aligned}$$\end{document}

which gives

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q(J)-Q(I)\leqq&-\big (1-C(L(I)+\sigma ^*-\sigma _*)\big )Q(\Lambda ). \end{aligned}$$\end{document}

When L(I) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*-\sigma _*$$\end{document} are small enough, and K is sufficiently large, it follows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(J)-F(I)\leqq -\left\{ K\big (1-C(L(I)+\sigma ^*-\sigma _*)\big )-C\right\} Q(\Lambda )\leqq -\frac{1}{4}Q(\Lambda ). \end{aligned}$$\end{document}

Case 2.  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _b=\Delta _{h,n_{b,h}-1}\cup \Delta _{h,n_{b,h}}\cup \Delta _{h,n_{b,h}+1}$$\end{document} covers a part of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta }$$\end{document} but none of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x, \vartheta }$$\end{document} . Direct computation shows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&L_{0}^{(1)}(J)-L_{0}^{(1)}(I)\leqq |K_{r,1}||\alpha _{r,2}| +|K_{\sigma ,1}||\Delta \sigma _{b_{h}}|+|K_{b,1}||\omega _{h+1}|+CQ(\Lambda _b),\\&L_{0}^{(2)}(J)-L_{0}^{(2)}(I)\leqq -|\alpha _{r,2}|+|K_{r,2}||\alpha _{r,2}| +|K_{\sigma ,2}||\Delta \sigma _{b_{h}}|\\&\qquad \qquad \qquad \qquad +|K_{b,2}||\omega _{h+1}|+CQ(\Lambda _b),\\&L_{1}(J)-L_{1}(I)=-|\omega _{h+1}|,\\&L_s(J)-L_s(I)=0,\\&L_b(J)-L_b(I)=-|\Delta \sigma _{b_{h}}|+|K_{c,2}||\alpha _{r,2}|+C|\omega _{h+1}|+C|\alpha _{r,1}||\Delta \sigma _{\alpha }|. \end{aligned}$$\end{document}

Combining the above estimates together, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(J)-L(I)&\leqq -\big (K_{2}-|K_{r,1}|-K_4|K_{c,2}|-K_{2}|K_{r,2}|\big )|\alpha _{r,2}|\\&\quad -\big (K_{1}-|K_{b,1}|-K_{2}|K_{b,2}|-CK_4\big )|\omega _{h+1}|\\&\quad -\big (K_4-|K_{\sigma ,1}|-K_{2}|K_{\sigma ,2}|\big )|\Delta \sigma _{b_{h}}|+CQ(\Lambda _b)+C|\alpha _{r,1}||\Delta \sigma _{\alpha }|. \end{aligned}$$\end{document}

For the terms contained in Q, noting that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\Delta \sigma _{\alpha }|\leqq |\Delta \sigma _{\nu }|$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q_{0}(J)-Q_{0}(I)&\leqq -Q^0((\alpha _{1},0),\nu )+CL(I)\big (|\alpha _{r,2}| +|\Delta \sigma _{b_{h}}|\\&\quad +|\omega _{h+1}|+CQ(\Lambda _b)\big ),\\ Q_{1}(J)-Q_{1}(I)&=|\delta _{1}|(\sigma _{\nu }-\sigma _*)-(|\alpha _{l,1}|+|\alpha _{r,1}|)(\sigma _{\alpha }-\sigma _*)-|\nu _{1}|(\sigma _{\nu }-\sigma _*)\\&\leqq -(|\alpha _{l,1}|+|\alpha _{r,1}|)|\Delta \sigma _{\nu }|+C(\sigma ^*-\sigma _*)\big (|\alpha _{r,2}|\\&\quad +|\Delta \sigma _{b_{h}}|+|\omega _{h+1}|+CQ(\Lambda _b)\big ),\\ Q_{2}(J)-Q_{2}(I)&=|\delta _{2}|(\sigma ^*-\sigma _{\nu })-|\alpha _{r,2}|(\sigma ^*-\sigma _{\alpha })-|\nu _{2}|(\sigma ^*-\sigma _{\nu })\\&\leqq C(\sigma ^*-\sigma _*)\big (|\alpha _{r,2}| +|\Delta \sigma _{b_{h}}|+|\omega _{h+1}|+CQ(\Lambda _b)\big ). \end{aligned}$$\end{document}

Then we conclude

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&Q(J)-Q(I)\\&\quad \leqq -Q^0((\alpha _{1},0),\nu )+CL(I)\big (|\alpha _{r,2}| +|\Delta \sigma _{b_{h}}|+|\omega _{h+1}|+CQ(\Lambda _b)\big )-|\alpha _{1}||\Delta \sigma _{\nu }|\\&\quad \quad -|\alpha _{r,1}||\Delta \sigma _{\alpha }|+2C(\sigma ^*-\sigma _*)\big (|\alpha _{r,2}| +|\Delta \sigma _{b_{h}}|+|\omega _{h+1}|+CQ(\Lambda _b)\big )\\&\quad \leqq -\big (1-C(L(I)+\sigma ^*-\sigma _*)\big )Q(\Lambda _b)\\&\quad \quad +C\big (L(I)+\sigma ^*-\sigma _*\big )\big (|\alpha _{r,2}| +|\Delta \sigma _{b_{h}}|+|\omega _{h+1}|\big ). \end{aligned}$$\end{document}

Finally, combining all the estimates above together, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&F(J)-F(I)\\&\quad \leqq -\left\{ K\big (1-C(L(I)+\sigma ^*-\sigma _*)\big )- C\right\} Q(\Lambda _b)\\&\quad \quad -\left\{ K_{2}-|K_{r,1}|-K_4|K_{c,2}|-K_{2}|K_{r,2}|-KC\big (L(I)+\sigma ^*-\sigma _*\big )\right\} |\alpha _{r,2}|\\&\quad \quad -\left\{ K_{1}-|K_{b,1}|-K_{2}|K_{b,2}|-CK_4-KC\big (L(I)+\sigma ^*-\sigma _*\big )\right\} |\omega _{h+1}|\\&\quad \quad -\left\{ K_4-|K_{\sigma ,1}|-K_{2}|K_{\sigma ,2}|-KC\big (L(I)+\sigma ^*-\sigma _*\big )\right\} |\Delta \sigma _{b_{h}}|. \end{aligned}$$\end{document}

Taking suitably large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{1}$$\end{document} , then, when K is sufficiently large, and L(I) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*-\sigma _*$$\end{document} are sufficiently small, we conclude

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(J)-F(I)\leqq -\frac{\xi }{4}\big (|\alpha _{r,2}|+|\omega _{h+1}|+|\Delta \sigma _{b_{h}}|+Q(\Lambda _b)\big ) \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi >0$$\end{document} small enough.

Case 3.  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _{s}=\Delta _{h,n_{\chi ,h}-1}\cup \Delta _{h,n_{\chi ,h}}\cup \Delta _{h,n_{\chi ,h}+1}$$\end{document} covers a part of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\Delta x,\vartheta }$$\end{document} but none of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x, \vartheta }$$\end{document} . A direct computation shows that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&L_{0}^{(1)}(J)-L_{0}^{(1)}(I)\leqq -|\alpha _{l,1}|+|K_{w,1}||\alpha _{l,1}|+|\mu _{w,1}||\Delta \sigma _{\chi _{h}}|+CQ(\Lambda _{s}),\\&L_{0}^{(2)}(J)-L_{0}^{(2)}(I) \leqq |K_{w,2}||\alpha _{l,1}|+|\mu _{w,2}||\Delta \sigma _{\chi _{h}}|+CQ(\Lambda _{s}),\\&L_{1}(J)-L_{1}(I)=0,\\&L_s(J)-L_s(I)\leqq -|\Delta \sigma _{\chi _{h}}|+|K_s||\alpha _{l,1}|,\\&L_b(J)-L_b(I)=0. \end{aligned}$$\end{document}

Combine the above estimates together, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&L(J)-L(I)\\&\quad \leqq -(1-|K_{w,1}|-K_{2}|K_{w,2}|-K_3|K_s|)|\alpha _{l,1}|\\&\quad \quad -\big (K_3-|\mu _{w,1}|-K_{2}|\mu _{w,2}|\big )|\Delta \sigma _{\chi _{h}}|+CQ(\Lambda _{s})\\&\quad \leqq -\big (1-K_{2}|K_{w,2}|-K_3|K_s|-|K_{w,1}|\big )|\alpha _{l,1}|\\&\quad \quad -\big (K_3-K_{2}|\mu _{w,2}|-|\mu _{w,1}|\big )|\Delta \sigma _{\chi _{h}}|+CQ(\Lambda _{s}). \end{aligned}$$\end{document}

For the terms contained in Q, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q_{0}(J)-Q_{0}(I)&\leqq -Q^0(\alpha _{r},\nu ) +CL(I)\big (|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|+Q(\Lambda _{s})\big ),\\ Q_{1}(J)-Q_{1}(I)&= |\delta _{1}|(\sigma _{\alpha }-\sigma _*)-(|\alpha _{l,1}|+|\alpha _{r,1}|)(\sigma _{\alpha }-\sigma _*)-|\nu _{1}|(\sigma _{\nu }-\sigma _*)\\&\leqq -|\nu _{1}||\Delta \sigma _{\nu }|+C(\sigma ^*-\sigma _*)\big (|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|+Q(\Lambda _{s})\big ),\\ Q_{2}(J)-Q_{2}(I)&= |\delta _{2}|(\sigma ^*-\sigma _{\nu })-|\alpha _{r,2}|(\sigma ^*-\sigma _{\alpha })\\&\leqq C(\sigma ^*-\sigma _*)\big (|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|+Q(\Lambda _{s})\big ). \end{aligned}$$\end{document}

Then we deduce that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q(J)-Q(I)\leqq&-\big (1-C(L(I)+\sigma ^*-\sigma _*)\big )Q(\Lambda _{s})\\&+C\big (L(I)+\sigma ^*-\sigma _*\big )\big (|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|\big ). \end{aligned}$$\end{document}

Finally, combining all the estimates above together, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&F(J)-F(I)\\&\quad \leqq -\left\{ K\big (1-C(L(I)+\sigma ^*-\sigma _*)\big ) -C\right\} Q(\Lambda _{s})\\&\quad \quad -\Big \{1-K_{2}|K_{w,2}|-K_3|K_s|-|K_{w,1}|-CK\big (L(I)+\sigma ^*-\sigma _*\big )\Big \}|\alpha _{l,1}|\\&\quad \quad -\Big \{K_3-K_{2}|\mu _{w,2}|-|\mu _{w,1}|-CK\big (L(I)+\sigma ^*-\sigma _*\big )\Big \}|\Delta \sigma _{\chi _{h}}|. \end{aligned}$$\end{document}

When K is sufficiently large, and L(I) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*-\sigma _*$$\end{document} are sufficiently small, we conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(J)-F(I)\leqq&-\frac{\xi }{4}(Q(\Lambda _{s})+|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|) \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi >0$$\end{document} small enough. Combining the above three cases, we conclude our result. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Now, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{h}$$\end{document} be the mesh curve in the stripe: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(x,y):\, x_{h-1}\leqq x\leqq x_{h}\}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in {\mathbb {N}}_{+}$$\end{document} ; that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{h}$$\end{document} connects all the mesh points in the strip. Let I and J be any pair of mesh curves with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{h}<I<J<I_{h+1}$$\end{document} , and let J be an immediate successor of I. That is, the mesh points on J differ from those on I by only one point generally (except three points near the approximate boundary or near the approximate shock), and the region bounded by the difference between I and J is denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} . Proposition 5.1 suggests that the total variation of the approximate solutions is uniformly bounded.

Moreover, we have the following estimates for the approximate boundary and the approximate leading shock:

Proposition 5.2

There exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{C}}>0$$\end{document} , independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x, \vartheta }$$\end{document} , such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\text {T.V.}\{s_{\Delta x,\vartheta }:[0,\infty )\}=\sum _{h=0}^{\infty }|s_{h+1}-s_{h}|\leqq {\bar{C}}\sum _{h\geqq 1}|\omega _{h}|,\\&\text {T.V.}\{b'_{\Delta x,\vartheta }:[0,\infty )\}=\sum _{h=0}^{\infty }|b'_{h+1}-b'_{h}|\leqq {\bar{C}}\sum _{h\geqq 1}|\omega _{h}|. \end{aligned}$$\end{document}

Proof

Notice that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {T.V.}\{s_{\Delta x,\vartheta }:[0,\infty )\}&=\sum _{h=0}^{\infty }|s_{h+1}-s_{h}|\leqq O(1)\sum _{\Lambda _s}E_{\Delta x,\vartheta }(\Lambda _s)\\&\leqq O(1)\sum _{\Lambda }F(I)-F(J)\leqq O(1)F(I_{1}). \end{aligned}$$\end{document}

Similarly, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {T.V.}\{b'_{\Delta x,\vartheta }:[0,\infty )\}\leqq O(1)F(I_{1}). \end{aligned}$$\end{document}

Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{C}}$$\end{document} in the statement can be determined. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

We choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{1}=2{\bar{C}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varpi =2{\bar{C}}\sum _{h\geqq 1}|\omega _{h}|$$\end{document} in (5.1). The largeness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} and the smallness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{h\geqq 1}|\omega _{h}|$$\end{document} imply the smallness of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^{*}-\sigma _{*}$$\end{document} . Then, following [14, 52], we conclude

Theorem 5.1

Under assumptions (A1)–(A2), if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} is sufficiently large and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{h\geqq 1}|\omega _{h}|$$\end{document} is sufficiently small, then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in \Pi _{h=0}^{\infty }[0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x>0$$\end{document} , the modified Glimm scheme introduced above defines a sequence of global approximate solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }(x,y)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{x>0}\text {T.V.}\{U_{\Delta x,\vartheta }(x,y):(-\infty ,b_{\Delta x,\vartheta }(x))\}<\infty ,\\&\int _{-\infty }^0|U_{\Delta x,\vartheta }(x_{1},y+b_{\Delta x,\vartheta }(x_{1}))-U_{\Delta x,\vartheta }(x_{2},y+b_{\Delta x,\vartheta }(x_{2}))|\,{\textrm{d}}y< L_{1}|x_{1}-x_{2}| \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{1}>0$$\end{document} independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} .

Convergence of the Approximate Solutions

In Section 5, the uniform bound of the total variation of the approximate solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} has been obtained. Then, by Propositions 5.1–5.2, the existence of convergent subsequences of the approximate solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{U_{\Delta x,\vartheta }\}$$\end{document} follows. Now we are going to prove that there is a convergent subsequence of the approximate solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{U_{\Delta x,\vartheta }\}$$\end{document} whose limit is an entropy solution to our problem.

Take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x=2^{-m}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m=0,1,2,\cdots $$\end{document} . For any randomly chosen sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta =(\vartheta _{0},\vartheta _{1},\vartheta _{2},\cdots ,\vartheta _{h},\cdots )$$\end{document} , we obtain a set of approximate solutions, which are denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(u_m,v_m)\}$$\end{document} . It suffices to prove that there is a subsequence (still denoted by) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(u_m,v_m)\}$$\end{document} such that, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\rightarrow \infty $$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\iint _{\Omega _{\Delta x,\vartheta }}\Big (\phi _x\rho _m u_m+\phi _y\rho _m v_m -\frac{\rho _m v_m\phi }{y}\Big )\,{\textrm{d}}x {\textrm{d}}y \nonumber \\&\qquad +\int _{-\infty }^{y_{0}(0)}\phi (x_{0},y)\rho (x_{0},y)u(x_{0},y)\,{\textrm{d}}y\rightarrow 0 \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (x,y)\in \text {C}_{0}^1({\mathbb {R}}^2;{\mathbb {R}})$$\end{document} , and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \iint _{\Omega _{\Delta x,\vartheta }}\left( \phi _x v_m-\phi _y u_m\right) {\textrm{d}}x {\textrm{d}}y\rightarrow 0 \end{aligned}$$\end{document}

for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (x,y)\in \text {C}_{0}^1(\Omega ;{\mathbb {R}})$$\end{document} . We now prove (6.1) only, since (6.2) can be deduced analogously.

For simplicity, we drop the subscript of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(u_m,v_m)$$\end{document} , and rewrite (6.1) as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\iint _{\Omega _{\Delta x,\vartheta }} \Big (\phi _x\rho u+\phi _y\rho v-\frac{\rho v\phi }{y}\Big )\,{\textrm{d}}x {\textrm{d}}y +\int _{-\infty }^{y_{0}(0)}\phi (x_{0},y)\rho (x_{0},y)u(x_{0},y)\,{\textrm{d}}y\\&\quad =\sum _{h=1}^{\infty }\iint _{\Omega _{\Delta x,\vartheta ,h}} \Big (\phi _x\rho u+\phi _y\rho v-\frac{\rho v\phi }{y}\Big )\,{\textrm{d}}x {\textrm{d}}y \\&\qquad +\int _{-\infty }^{y_{0}(0)}\phi (x_{0},y)\rho (x_{0},y)u(x_{0},y)\, {\textrm{d}}y. \end{aligned}$$\end{document}

By the shock waves and the upper/lower edges of rarefaction waves, each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h}$$\end{document} can be divided into smaller polygons: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h,j}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=0,-1,-2,\cdots $$\end{document} , alternatively, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h,0}$$\end{document} is the uppermost area below the approximate boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\Delta x,\vartheta ,h}$$\end{document} . Then we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\iint _{\Omega _{\Delta x,\vartheta }} \Big (\phi _x\rho u+\phi _y\rho v-\frac{\rho v\phi }{y}\Big )\,{\textrm{d}}x{\textrm{d}}y +\int _{-\infty }^{y_{0}(0)}\phi (x_{0},y)\rho (x_{0},y)u(x_{0},y)\,{\textrm{d}}y\\&\quad =\sum _{h=1}^{\infty }\sum _{j=0}^{-\infty }\iint _{\Omega _{\Delta x,\vartheta ,h,j}} \Big (\phi _x\rho u+\phi _y\rho v-\frac{\rho v\phi }{y}\Big )\,{\textrm{d}}x {\textrm{d}}y\\&\quad \quad +\int _{-\infty }^{y_{0}(0)}\phi (x_{0},y)\rho (x_{0},y)u(x_{0},y)\,{\textrm{d}}y\\&\quad =-\sum _{h,j}\iint _{\Omega _{\Delta x,\vartheta ,h,j}} \phi \Big ((\rho u)_x+(\rho v)_y+\frac{\rho v}{y}\Big )\,{\textrm{d}}x {\textrm{d}}y\\&\quad \quad +\sum _{h,j}\iint _{\Omega _{\Delta x,\vartheta ,h,j}} \big ( (\phi \rho u)_x+(\phi \rho v)_y\big )\,{\textrm{d}}x {\textrm{d}}y\\&\quad \quad +\int _{-\infty }^{y_{0}(0)}\phi (x_{0},y)\rho (x_{0},y)u(x_{0},y)\,{\textrm{d}}y\\&\quad =:\text {I}+\text {II} +\text {III}. \end{aligned}$$\end{document}

We first have

Proposition 6.1

I \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow 0\,\,$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x\rightarrow 0$$\end{document} .

Proof

To deal with the first term I, we use the transform:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma =\frac{y}{x},\qquad \eta =\frac{y-y_n(h)}{x-x_{h}}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},y_n(h))$$\end{document} is the center of the Riemann problem, and n depends on j. Then we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \text {I} =&\sum _{h,j}\iint \frac{(x-x_h)\phi \rho }{\sigma (\eta -\sigma )} \Big (\sigma ^2\big (1-\frac{u^2}{c^2}\big ) u_{\sigma }-\frac{2uv\sigma ^2}{c^2}v_{\sigma }-\big (1-\frac{v^2}{c^2}\big ) v_{\sigma }\sigma -v\Big )\,{\textrm{d}}\eta {\textrm{d}}\sigma \\&-\sum _{h,j}\iint \frac{x\phi }{\eta -\sigma } \big (-\eta (\rho u)_{\eta }+(\rho v)_{\eta }\big )\,{\textrm{d}}\eta {\textrm{d}}\sigma . \end{aligned} \end{aligned}$$\end{document}

From the construction of the approximate solutions, the first term of (6.4) vanishes. For the second term, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\eta (\rho u)_{\eta }+(\rho v)_{\eta }=O(1)\Delta \sigma , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma $$\end{document} is the change of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma -$$\end{document} coordinate in domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h,j}$$\end{document} . Denote the rarefaction waves in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h}$$\end{document} alternatively by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{R,h,i}$$\end{document} . Then we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {I}=O(1)\sum _{h,j}\Delta \eta (\Delta \sigma )^2, \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \eta =O(1)\alpha _{R,h,i}$$\end{document} . According to Proposition 5.1, the total strength \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i}|\alpha _{R,h,i}|$$\end{document} of rarefaction waves in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h}$$\end{document} is bounded, so that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {I}=O(1)\,\text {diam}(\text {supp}\,\phi )\,\Delta x, \end{aligned}$$\end{document}

which gives desired result. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Next, applying Green’s formula in each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h,j}$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \text {II}+\text {III}&=\sum _{h=1}^{\infty }\int _{-\infty }^{b_{\Delta x,\vartheta }(x_{h})}\phi (x_{h},y)\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad \, +\sum _{h=0}^{\infty }\int _{x_{h}}^{x_{h+1}}\phi (x,b(x)) \rho (x,b(x)) \big (v(x,b(x))-u(x,b(x))b'(x)\big ){\textrm{d}}x\\&\quad \, +\sum _{h,i}\int _{W_{h,i}} \big (s_{h,i}(\rho ^+u^+-\rho ^-u^-)-(\rho ^+v^+-\rho ^-v^-)\big )\phi \,{\textrm{d}}x\\&=: \text {IV}+\text {V}+\text {VI}, \end{aligned} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{h,i}=\{(x,y):\,y=w_{h,i}(x)=s_{h,i}(x-x_{h})+y_n(h)\,\text { for some }n\}$$\end{document} are shock waves or upper/lower edges of rarefaction waves lying in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^\pm =\rho (x,w_{i,h}(x)\pm )$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^\pm =u(x,w_{i,h}(x)\pm )$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^\pm =v(x,w_{i,h}(x)\pm )$$\end{document} .

We now show

Proposition 6.2

There exists a subsequence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(u_m,v_m)\}$$\end{document} such that IV \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow 0\,\,$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\rightarrow \infty $$\end{document} .

Proof

The first term on the right-hand side of (6.6) can be rewritten as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {IV}=\sum _{h\geqq 1}V_{h} \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} V_{h}&=\sum _{n=n_{\chi ,h}+1}^{n_{b,h}-1}\int _{y_{n-1}(h)}^{y_{n}(h)}\phi (x_{h},y)\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad \,+\int _{\chi _{\Delta x,\vartheta }(x_{h})}^{y_{n_{\chi ,h}}(h)+1}\phi (x_{h},y)\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad \,+\int _{y_{n_{b,h}}(h)-1}^{b_{\Delta x,\vartheta }(x_{h})}\phi (x_{h},y)\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y. \end{aligned}$$\end{document}

To show \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {IV}\rightarrow 0$$\end{document} for some subsequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(u_m,v_m)\}$$\end{document} , we now introduce

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widetilde{V}}=\sum _{h\geqq 1}{\widetilde{V}}_{h} \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widetilde{V}}_{h}&=\sum _{n=n_{\chi ,h}+1}^{n_{b,h}-1}\int _{y_{n-1}(h)}^{y_{n}(h)}\phi (x_{h},y_n(h))\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad \,+\int _{y_{n_{b,h}}(h)}^{b_{\Delta x,\vartheta }(x_{h})}\phi (x_{h},y_{n_{b,h}+1}(h))\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad \,+\int _{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi (x_{h},y_{n_{b,h}}(h))\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big ){\textrm{d}}y\\&\quad \,+\int _{y_{n_{\chi ,h}}(h)}^{y_{n_{\chi ,h}+1}(h)}\phi (x_{h},y_{n_{\chi ,h}+1}(h))\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad \,+\int _{\chi _{\Delta x,\vartheta }(x_{h})}^{y_{n_{\chi ,h}}(h)}\phi (x_{h},y_{n_{\chi ,h}}(h))\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&=:\sum _{n=n_{\chi ,h}+1}^{n_{b,h}-1}\int _{y_{n-1}(h)}^{y_{n}(h)}\phi (x_{h},y_n(h))\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad \,+{\check{V}}_{h}^{(0)}+\check{V}_{h}^{(1)}+{\hat{V}}_{h}^{(1)}+{\hat{V}}_{h}^{(0)}. \end{aligned}$$\end{document}

From the construction of approximate solutions, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \check{V}_{h}^{(0)}&=O(1)\Delta x\big (|\alpha _{r,2}|+|\omega _{h+1}|+|\Delta \sigma _{b_{h}}|+Q(\Lambda _b)\big ) & (\text {see Case }2),&\\ {\hat{V}}_{h}^{(0)}&=O(1)\Delta x\big (|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|+Q(\Lambda _{s})\big ) & (\text {see Case }3).&\end{aligned}$$\end{document}

From Proposition 5.1, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{h=1}^{\infty }\big (\check{V}_{h}^{(0)}+{\hat{V}}_{h}^{(0)}\big )=O(1)\Delta xF(I_{1}). \end{aligned}$$\end{document}

Then we write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\check{V}_{h}^{(1)}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{V}}_{h}^{(1)}$$\end{document} as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \check{V}_{h}^{(1)}&=\int _{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)} \phi (x_{h},y_{n_{b,h}}(h))\\ &\quad \times \big (\rho (x_{h}-,y)u(x_{h}-,y) -{\check{\rho }}(x_{h}+,y)\check{u}(x_{h}+,y)\big ) \,{\textrm{d}}y\\&\quad +\int _{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)} \phi (x_{h},y_{n_{b,h}}(h))\\&\quad \times \big ({\check{\rho }}(x_{h}+,y)\check{u}(x_{h}+,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y \end{aligned}$$\end{document}

and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\hat{V}}_{h}^{(1)}&=\int _{y_{n_{\chi ,h}}(h)}^{y_{n_{\chi ,h}+1}(h)} \phi (x_{h},y_{n_{\chi ,h}}(h))\\ &\quad \times \big (\rho (x_{h}-,y)u(x_{h}-,y) -{\hat{\rho }}(x_{h}+,y){\hat{u}}(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad +\int _{y_{n_{\chi ,h}}(h)}^{y_{n_{\chi ,h}+1}(h)} \phi (x_{h},y_{n_{\chi ,h}}(h))\\ &\quad \times \big ({\hat{\rho }}(x_{h}+,y){\hat{u}}(x_{h}+,y) -\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y, \end{aligned}$$\end{document}

where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\check{U}(x_{h}+,y)= {\tilde{U}}(\dfrac{y}{x_{h}};\dfrac{r_{h,n_{b,h}-1}}{x_{h}},U(x_{h}+,r_{h,n_{b,h}-1})),\\&{\hat{U}}(x_{h}+,y)={\tilde{U}}(\dfrac{y}{x_{h}};\dfrac{r_{h,n_{\chi ,h}-1}}{x_{h}},U(x_{h}+,r_{h,n_{\chi ,h}-1})), \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\check{\rho }}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\rho }}$$\end{document} are determined via Bernoulli’s equation. By the construction of the approximate solutions near the boundary and near the leading shock, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi \big (x_{h},y_{n_{b,h}}(h)\big )\big ({\check{\rho }}(x_{h}+,y)\check{u}(x_{h}+,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big ){\textrm{d}}y\\&\,\,\,\,\, =O(1)\Delta x\big (|\alpha _{r,2}|+|\omega _{h+1}|+|\Delta \sigma _{b_{h}}|+Q(\Lambda _b)\big )\quad (\text {see Case }2),&\\&\int _{y_{n_{\chi ,h}}(h)}^{y_{n_{\chi ,h}+1}(h)}\phi \big (x_{h},y_{n_{\chi ,h}}(h)\big )\big ({\hat{\rho }}(x_{h}+,y){\hat{u}}(x_{h}+,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big ){\textrm{d}}y\\&\,\,\,\,\, =O(1)\Delta x\big (|\alpha _{l,1}|+|\Delta \sigma _{\chi _{h}}|+Q(\Lambda _{s})\big ) \quad (\text {see Case }3).&\end{aligned}$$\end{document}

Similarly, by Proposition 5.1, we conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\sum _{h=1}^{\infty }\int _{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi (x_{h},y_{n_{b,h}}(h))\big ({\check{\rho }}(x_{h}+,y)\check{u}(x_{h}+,y) -\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\,\,\,=O(1)\Delta xF(I_{1}),\\&\sum _{h=1}^{\infty }\int _{y_{n_{\chi ,h}}(h)}^{y_{n_{\chi ,h}+1}(h)} \phi (x_{h},y_{n_{\chi ,h}}(h))\big ({\hat{\rho }}(x_{h}+,y){\hat{u}}(x_{h}+,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\,\,\,=O(1)\Delta xF(I_{1}). \end{aligned} \end{aligned}$$\end{document}

Set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\bar{V}}_{h}&= \sum _{n=n_{\chi ,h}+1}^{n_{b,h}-1}\int _{y_{n-1}(h)}^{y_{n}(h)} \phi (x_{h},y_n(h))\big (\rho (x_{h}-,y)u(x_{h}-,y) -\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad +\int _{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)} \phi (x_{h},y_{n_{b,h}}(h)) \big (\rho (x_{h}-,y)u(x_{h}-,y) -{\check{\rho }}(x_{h}+,y)\check{u}(x_{h}+,y)\big )\,{\textrm{d}}y\\&\quad +\int _{y_{n_{\chi ,h}}(h)}^{y_{n_{\chi ,h}+1}(h)} \phi (x_{h},y_{n_{\chi ,h}}(h))\big (\rho (x_{h}-,y)u(x_{h}-,y) -{\hat{\rho }}(x_{h}+,y){\hat{u}}(x_{h}+,y)\big )\,{\textrm{d}}y. \end{aligned}$$\end{document}

As in [24] (see also [18]), let

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H=\prod _{h=0}^{\infty }[0,1) =\{\vartheta =(\vartheta _{0},\vartheta _{1},\vartheta _{2},\cdots , \vartheta _{h},\cdots ):\,\vartheta _{h}\in [0,1),\ h=0,1,2,\cdots \}. \end{aligned}$$\end{document}

Denoting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{y}}=y_{n-1}(h)+\vartheta _{h}\big (y_{n}(h)-y_{n-1}(h)\big )$$\end{document} , we obtain from (4.2) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\\&\quad =\rho (x_{h}-,y)u(x_{h}-,y) -\rho (x_{h}-,{\bar{y}})u(x_{h}-,{\bar{y}})+\rho (x_{h}+,{\bar{y}})u(x_{h}+,{\bar{y}})\\&\qquad -\rho (x_{h}+,y)u(x_{h}+,y)\\&\quad =O(1)|\alpha |+O(1)|\alpha ||\Delta \sigma _\alpha |+O(1)|\Delta \sigma _{\alpha }|\\&\quad =O(1)(|\alpha |+|\Delta \sigma _\alpha |), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is an elementary wave in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h,j}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \sigma _{\alpha }$$\end{document} is the change of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma -$$\end{document} coordinate in the elementary wave \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} . Denote the elementary waves in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h}$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{h,i}$$\end{document} . Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\bar{V}}_{h}=O(1)\Big (\sum _{i\leqq 0}|\alpha _{h,i}|+\sigma ^*-\sigma _*\Big )\Delta x, \end{aligned}$$\end{document}

which implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{h\geqq 1}\int _{H}{\bar{V}}_{h}^2{\textrm{d}}\vartheta =O(1)\text {diam}(\text {supp}\,\phi )\Big (\sum _{i\leqq 0}|\alpha _{h,i}|+\sigma ^*-\sigma _*\Big )^2\Delta x. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Next, we need the following lemma:

Lemma 6.1

The approximate solutions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{U_{\Delta x,\vartheta }(x,y)\}$$\end{document} satisfy

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\int _{0}^1\int _{y_{n-1}(h)}^{y_n(h)} \big (U_{\Delta x,\vartheta }(x_{h}-,y)-U_{\Delta x,\vartheta }(x_{h}+,y)\big )\, {\textrm{d}}y {\textrm{d}}\vartheta _{h}\\&\quad =O(1)(\Delta x)^3+O(1)(|\alpha |+|\beta |)(\Delta x)^2. \end{aligned} \end{aligned}$$\end{document}

Proof

We now give a proof when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} are both shock waves, since the remaining cases can be obtained similarly.

Suppose that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} issue from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n-1}(h-1))$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h-1},y_{n}(h-1))$$\end{document} , and end at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},r_{1})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{h},r_{2})$$\end{document} , respectively. Set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{1}=\frac{r_{1}-y_{n-1}(h)}{y_{n}(h)-y_{n-1}(h)}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a_{2}=\frac{r_{2}-y_{n-1}(h)}{y_{n}(h)-y_{n-1}(h)}$$\end{document} . From the construction of approximate solutions, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{0}^1\int _{y_{n-1}(h)}^{y_n(h)} \big (U_{\Delta x,\vartheta }(x_{h}-,y)-U_{\Delta x,\vartheta }(x_{h}+,y)\big )\,{\textrm{d}}y {\textrm{d}}\vartheta _{h}\\&\quad =\int _{0}^1\int _{y_{n-1}(h)}^{y_n(h)}U_{\Delta x,\vartheta }(x_{h}-,y)\,{\textrm{d}}y {\textrm{d}}\vartheta _{h}- \int _{0}^1\int _{y_{n-1}(h)}^{y_n(h)}U_{\Delta x,\vartheta }(x_{h}+,y)\,{\textrm{d}}y {\textrm{d}}\vartheta _{h}\\&\quad =\int _{y_{n-1}(h)}^{r_{1}}{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},U_{l})\,{\textrm{d}}y +\int _{r_{1}}^{r_{2}}{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))\,{\textrm{d}}y\\&\quad \quad +\int _{r_{2}}^{y_n(h)}{\tilde{U}}(\frac{y}{x_{h}};\sigma _{1},\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))))\,{\textrm{d}}y\\&\quad \quad -a_{1}\int _{y_{n-1}(h)}^{y_n(h)}{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},U_{l})\,{\textrm{d}}y\\&\quad \quad -(a_{2}-a_{1})\int _{y_{n-1}(h)}^{y_n(h)}{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))\,{\textrm{d}}y\\&\quad \quad -(1-a_{2})\int _{y_{n-1}(h)}^{y_n(h)}{\tilde{U}}(\frac{y}{x_{h}};\sigma _{1},\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))))\,{\textrm{d}}y. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l})) ={\tilde{U}}(\frac{y}{x_{h}};{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))$$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{0}^1\int _{y_{n-1}(h)}^{y_n(h)}(U_{\Delta x,\vartheta }(x_{h}-,y)-U_{\Delta x,\vartheta }(x_{h}+,y))\,{\textrm{d}}y {\textrm{d}}\vartheta _{h}\\&\quad =\int _{y_{n-1}(h)}^{r_{1}}(1-a_{1})\big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},U_{l})-{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))\big )\,{\textrm{d}}y\\&\quad \quad -\int _{y_{n-1}(h)}^{r_{1}}(1-a_{2})\big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{1},\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))))\\&\qquad \qquad \qquad -{\tilde{U}}(\frac{y}{x_{h}};{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))\big )\,{\textrm{d}}y\\&\quad \quad +\int _{r_{1}}^{r_{2}}a_{1}\big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))-{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},U_{l})\big )\,{\textrm{d}}y\\&\quad \quad -\int _{r_{1}}^{r_{2}}(1-a_{2})\big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{1},\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))))\\&\qquad \qquad \qquad -{\tilde{U}}(\frac{y}{x_{h}};{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))\big )\,{\textrm{d}}y\\&\quad \quad +\int _{r_{2}}^{y_n(h)}a_{2}\big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{1},\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))))\\&\qquad \qquad \qquad -{\tilde{U}}(\frac{y}{x_{h}};{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))\big )\, {\textrm{d}}y\\&\quad \quad -\int _{r_{2}}^{y_n(h)}a_{1}\big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},U_{l})-{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))\big )\,{\textrm{d}}y. \end{aligned}$$\end{document}

Then, by Taylor’s expansion, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},U_{l}) -{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))\\&\quad =U_{l}-\Phi (0,\alpha _{2};U_{l})+A_{1}(y-y_{n-1}(h))+O(1)(y-y_{n-1}(h))^2, \\&{\tilde{U}}(\frac{y}{x_{h}};\sigma _{1},\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))) -{\tilde{U}}(\frac{y}{x_{h}};{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))\\&\quad =\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))) -{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))\\&\quad \quad +A_{2}(y-y_{n}(h))+O(1)(y-y_{n}(h))^2, \end{aligned}$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{1}=&\left. \partial _y\Big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},U_{l})-{\tilde{U}}(\frac{y}{x_{h}};\sigma _{2},\Phi (0,\alpha _{2};U_{l}))\Big ) \right| _{y=y_{n-1}(h)},\\ A_{2}=&\left. \partial _y\Big ({\tilde{U}}(\frac{y}{x_{h}};\sigma _{1},\Phi (\beta _{1},0;{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))) \right. \\&\quad \left. -{\tilde{U}}(\frac{y}{x_{h}};{\tilde{U}}(\sigma _{1};\sigma _{2},\Phi (0,\alpha _{2};U_{l})))\Big ) \right| _{y=y_{n}(h)}. \end{aligned}$$\end{document}

A direct computation leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{0}^1\int _{y_{n-1}(h)}^{y_n(h)} \big (U_{\Delta x,\vartheta }(x_{h}-,y)-U_{\Delta x,\vartheta }(x_{h}+,y)\big )\,{\textrm{d}}y {\textrm{d}}\vartheta _{h}\\&\quad = O(1)(y_{n}(h)-y_{n-1}(h))^3+\dfrac{1}{2}A_{1}(r_{1}-y_{n-1}(h))(r_{1}-y_{n}(h))\\&\quad \quad -\dfrac{1}{2}A_{2}(r_{2}-y_{n-1}(h))(r_{2}-y_{n}(h)). \end{aligned}$$\end{document}

Noting that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{1}=O(1)|\alpha |$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{2}=O(1)|\beta |$$\end{document} , together with the Courant-Friedrichs-Lewy condition, we conclude (6.10). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Substituting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} in Lemma 6.1 by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho u$$\end{document} and carrying out the same process leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{0}^1{\bar{V}}_{h}{\textrm{d}}\vartheta _{h} =O(1)\Big (\text {diam}(\text {supp}\,\phi )+\sum _{i\leqq 0}|\alpha _{h,i}|\Big )(\Delta x)^2=O(1)(\Delta x)^2. \end{aligned}$$\end{document}

As in (6.9), we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\bar{V}}_k=O(1)\Big (\sum _{i\leqq 0}|\alpha _{k,i}|+\sigma ^*-\sigma _*\Big )\Delta x. \end{aligned}$$\end{document}

Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{h>k}\int _{H}{\bar{V}}_{h} {\bar{V}}_k\,{\textrm{d}}\vartheta \leqq&\sum _{h>k}\left| \int _{0}^1 {\bar{V}}_{h}d\vartheta _{h}\right| \int _{0}^1| {\bar{V}}_k|\,{\textrm{d}}{\hat{\vartheta }}_{h} =O(1)\big (\text {diam}(\text {supp,}\phi )\big )^2\Delta x, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{d}}{\hat{\vartheta }}_{h} ={\textrm{d}}\vartheta _{0}\cdots {\textrm{d}}\vartheta _{h-1}{\textrm{d}}\vartheta _{h+1}\cdots $$\end{document} .

Since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\bar{V}}\Vert _{L^2(H)} =\sum _{h\geqq 1}\int _{H}{\bar{V}}_{h}^2{\textrm{d}}\vartheta +2\sum _{h>k}\int _{H}{\bar{V}}_{h}{\bar{V}}_k\,{\textrm{d}}\vartheta , \end{aligned}$$\end{document}

we conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\bar{V}}\Vert _{L^2(H)}\rightarrow 0\qquad \text {as }\Delta x\rightarrow 0, \end{aligned}$$\end{document}

which, combining with (6.7)–(6.8), gives a subsequence (still denoted by) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(u_m,v_m)\}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{V}}\rightarrow 0$$\end{document} almost everywhere. Meanwhile, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&V_{h}-{\widetilde{V}}_{h}\\ &\quad = \sum _{n=n_{\chi ,h}+1}^{n_{b,h}-1}\int _{y_{n-1}(h)}^{y_n(h)}\big (\phi (x_{h},y_n(h))-\phi (x_{h},y)\big )\\ &\qquad \qquad \qquad \qquad \,\,\,\times \big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\text {d}}y\\ &\quad \quad +\int _{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)} \big (\phi (x_{h},y_{n_{b,h}}(h))-\phi (x_{h},y)\big )\\ &\qquad \qquad \qquad \,\,\,\,\times \big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\text {d}}y\\ &\quad \quad +\int _{y_{n_{b,h}}(h)}^{b_{\Delta x,\vartheta }(x_{h})} \big (\phi (x_{h},y_{n_{b,h}+1}(h))-\phi (x_{h},y)\big )\\ &\qquad \qquad \qquad \quad \times \big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\text {d}}y\\ &\quad \quad +\int _{\chi _{\Delta x,\vartheta }(x_{h})}^{y_{n_{\chi ,h}}(h)} \big (\phi (x_{h},y_{n_{\chi ,h}}(h))-\phi (x_{h},y)\big )\\ &\qquad \qquad \qquad \,\,\,\,\times \big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\text {d}}y\\ &\quad \quad +\int _{y_{n_{\chi ,h}}(h)}^{y_{n_{\chi ,h}+1}(h)} \big (\phi (x_{h},y_{n_{\chi ,h}+1}(h))-\phi (x_{h},y)\big )\\ &\qquad \qquad \qquad \quad \,\times \big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\text {d}}y\\ &\quad =O(1)\Delta x\sum _{n=n_{\chi ,h}+1}^{n_{b,h}-1}\int _{y_{n-1}(h)}^{y_n(h)}\big (\rho (x_{h}-,y)u(x_{h}-,y)-\rho (x_{h}+,y)u(x_{h}+,y)\big )\,{\text {d}}y \\ &\qquad +O(1)(\Delta x)^2\\&\quad =O(1)\Big (\sum _{i\leqq 0}|\alpha _{h,i}|+\sigma ^*-\sigma _*+1\Big )(\Delta x)^2, \end{aligned} \end{aligned}$$\end{document}

which leads to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {IV}-{\widetilde{V}}=&\sum _{h\geqq 1}V_{h}-{\widetilde{V}}_{h} =O(1)\,\text {diam}(\text {supp}\,\phi )\Delta x. \end{aligned}$$\end{document}

Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {IV}\rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m\rightarrow \infty $$\end{document} for some subsequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(u_m,v_m)\}$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proposition 6.3

V, VI \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow 0$$\end{document}   as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x\rightarrow 0$$\end{document} .

Proof

Since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} b'(x)=\frac{v(x_{h}+,b(x_{h})-)}{u\big (x_{h}+,b(x_{h})-\big )}\qquad \text { for }x\in (x_{h},x_{h+1}), \end{aligned}$$\end{document}

it follows from the construction of our approximate solution that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} v\big (x,b(x)\big )-u\big (x,b(x)\big )b'(x)=O(1)\Delta x. \end{aligned}$$\end{document}

Therefore, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {V} =O(1)\,\text {diam}(\text {supp}\,\phi )\,\Delta x\rightarrow 0 \qquad \text {as }\Delta x\rightarrow 0. \end{aligned}$$\end{document}

As for VI, we divide this term into three parts. The first part is the integral along the leading shock, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{h,i}=S_{\Delta x,\vartheta ,h}$$\end{document} . For this part, by similar arguments in treating V, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{h}\int _{S_{\Delta x,\vartheta ,h}}\big (s_{h}(\rho ^+u^+-\rho ^-u^-)-(\rho ^+v^+-\rho ^-v^-)\big )\phi \,{\textrm{d}}x =O(1)\Delta x. \end{aligned}$$\end{document}

The second part is the integral along the upper or lower edges of rarefaction waves and therefore vanishes automatically. The third part is the integral along the weak shock waves, that is, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_{h,i}\ne S_{\Delta x,\vartheta ,h}$$\end{document} . In this case, by (4.1), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\rho ^+u^+-\rho ^-u^-=(\rho ^+u^+-\rho ^-u^-)|_{x=x_{h}+}+O(1)(\rho ^+u^+-\rho ^-u^-)|_{x=x_{h}+}\Delta x,\\&\rho ^+v^+-\rho ^-v^-=(\rho ^+v^+-\rho ^-v^-)|_{x=x_{h}+}+O(1)(\rho ^+v^+-\rho ^-v^-)|_{x=x_{h}+}\Delta x. \end{aligned}$$\end{document}

Thus, in view of the Rankine-Hugoniot conditions, we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{i}(s_{h,i}(\rho ^+u^+-\rho ^-u^-)-(\rho ^+v^+-\rho ^-v^-))=O(1)\sum _{i}|\alpha _{S,h,i}|\Delta x, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _{S,h,i}$$\end{document} are the weak shock waves in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{\Delta x,\vartheta ,h}$$\end{document} . Combining all the three parts together, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {IV} =O(1)\,\text {diam}(\text {supp}\,\phi )\,\sum _{i}|\alpha _{S,h,i}|\Delta x+O(1)\Delta x. \end{aligned}$$\end{document}

By Proposition 5.1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i}|\alpha _{S,h,i}|$$\end{document} is uniformly bounded with respect to h. Therefore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {IV}\rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x\rightarrow 0$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

With all the arguments stated above, a standard procedure as in [18, 38] gives the following theorem, which ensures the first part of the main theorem:

Theorem 6.1

Suppose that (A1)–(A2), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\gamma <3$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<p_{0}<p^*$$\end{document} hold. Then, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{\infty }$$\end{document} is sufficiently large, there are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0>0$$\end{document} , a null set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {N}}$$\end{document} , and a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} , depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0$$\end{document} and the system, such that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T.V.\,\{p^{b}\}=\varepsilon _p<\varepsilon _0$$\end{document} , for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in \prod _{h=0}^{\infty }[0,1)\backslash {\mathcal {N}}$$\end{document} , there exist both a subsequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\Delta _{i}\}_{i=0}^{\infty }\subset \{\Delta x\}$$\end{document} of the mesh size with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{i}\rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\rightarrow \infty $$\end{document} and a triple of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{\vartheta }(x)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{\vartheta }(0)=0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\vartheta }(x)$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\vartheta }(0)=0$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\vartheta }(x,y)\in O_{\varepsilon _{0}}\big (G(s_{0})\cap \mathbb {W}(p_{0},p_{\infty })\big )$$\end{document} such that

• (i) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{\Delta _{i},\vartheta }$$\end{document} converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{\vartheta }$$\end{document} uniformly in any bounded x-interval;
• (ii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\Delta _{i},\vartheta }$$\end{document} converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\vartheta }$$\end{document} uniformly in any bounded x-interval;
• (iii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_{\Delta _{i},\vartheta }$$\end{document} converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(b'_{\vartheta })_+\in BV([0,\infty ))$$\end{document} a.e., satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x>0}|(b'_{\vartheta })_+(x)-b_0|<C\varepsilon _p,\quad \quad b_{\vartheta }(x)=\int _0^x (b'_{\vartheta })_+(t){\textrm{d}}t\textrm{;} \end{aligned}$$\end{document}

• (iv) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{\Delta _{i},\vartheta }$$\end{document} converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{\vartheta }\in BV([0,\infty ))$$\end{document} a.e., satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x>0}|s_{\vartheta }(x)-s_0|<C\varepsilon _p,\quad \quad \chi _{\vartheta }(x)=\int _0^x s_{\vartheta }(t){\textrm{d}}t; \end{aligned}$$\end{document}

• (v) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta _{i},\vartheta }(x,\cdot )$$\end{document} converges to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\vartheta }\in {{\textbf {L}}}^1_{\textrm{loc}}(-\infty ,b_{\vartheta }(x))$$\end{document} for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>0$$\end{document} , so that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \sup _{x>0}\text{ T.V. }\{U_\vartheta (x,y)\, :\, \chi (x)<y<b(x)\}<C(\varepsilon _p+b_0-s_0), \end{aligned} \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\vartheta }$$\end{document} is a global entropy solution of the inverse problem (1.1)–(1.2) and satisfies (1.8)–(1.9).

Asymptotic Behavior of Global Entropy Solutions

To establish the asymptotic behavior of global entropy solutions, we need further estimates of the approximate solutions.

Lemma 7.1

There exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{1}$$\end{document} , independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta x$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} , such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\Lambda }E_{\Delta x,\vartheta }(\Lambda )<M_{1} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\Delta x,\vartheta }(\Lambda )$$\end{document} given as in (5.2).

Proof

By Proposition 5.1, for any interaction region \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \subset \{(h-1)\Delta x\leqq (h+1)\Delta x\}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\geqq 1$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{\Lambda }E_{\Delta x,\vartheta }(\Lambda )\leqq 4\sum _{\Lambda }\big (F(I)-F(J)\big )\leqq 4F(I_{1}). \end{aligned}$$\end{document}

Thus, choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_{1}=4F(I_{1})+1$$\end{document} , the proof is complete. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{j,\vartheta }(t-)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document} , be the total variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j-$$\end{document} weak waves in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\vartheta }$$\end{document} crossing line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=t$$\end{document} , and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{j,\Delta x, \vartheta }(t-)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document} , be the total variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j-$$\end{document} weak waves in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta x,\vartheta }$$\end{document} crossing line \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=t$$\end{document} . Then we have

Lemma 7.2

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{j=1}^{2}{\mathcal {L}}_{j,\vartheta }(x-)\rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\rightarrow \infty $$\end{document} .

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta _{i},\vartheta }$$\end{document} be a sequence of the approximate solutions introduced in Theorem 6.1, and let the corresponding term \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\Delta x,\vartheta }(\Lambda )$$\end{document} be defined in (5.2). As in [25], denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{d}}E_{\Delta x,\vartheta }$$\end{document} the measure of assigning quantities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\Delta x,\vartheta }(\Lambda )$$\end{document} to the center of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} . Then, by Lemma 7.1, we can choose a subsequence (still denoted as) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textrm{d}}E_{\Delta _{i},\vartheta }$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\textrm{d}}E_{\Delta _{i},\vartheta }\rightarrow dE_{\vartheta } \qquad \text {as } \Delta _{i}\rightarrow 0 \end{aligned}$$\end{document}

with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\vartheta }(\Lambda )<\infty $$\end{document} .

Therefore, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{1}>0$$\end{document} sufficiently small, we can choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{\varepsilon _{1}}$$\end{document} (independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta _{i},\vartheta }$$\end{document} ), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{i}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{h>[x_{\varepsilon _{1}}/\Delta x]} E_{\Delta _{i},\vartheta }(\Lambda _{h,n})<\varepsilon _{1}. \end{aligned}$$\end{document}

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\varepsilon _{1}}^{1}=(x_{\varepsilon _{1}},\chi _{\Delta _{i},\vartheta }(x_{\varepsilon _{1}}))$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\varepsilon _{1}}^{2}=(x_{\varepsilon _{1}},b_{\Delta _{i},\vartheta }(x_{\varepsilon _{1}}))$$\end{document} be the two points lying in the approximate leading shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=\chi _{\Delta _{i},\vartheta }(x)$$\end{document} and the approximate boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=b_{\Delta _{i},\vartheta }(x)$$\end{document} , respectively. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\Delta _{i},\vartheta }^{j}$$\end{document} be the approximate j–generalized characteristic issuing from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\varepsilon _{1}}^{j}$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document} , respectively. According to the construction of the approximate solutions, there exist constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{M}}_{j}>0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document} , independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\Delta _{i},\vartheta }$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{i}$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} , such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\chi _{\Delta _{i},\vartheta }^{j}(x_{1})-\chi _{\Delta _{i},\vartheta }^{j}(x_{2})|\leqq {\hat{M}}_{j}\big (|x_{1}-x_{2}|+\Delta _{i}\big ) \qquad \text {for } x_{1},x_{2}>x_{\varepsilon _{1}}. \end{aligned}$$\end{document}

Then we choose a subsequence (still denoted by) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{i}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \chi _{\Delta _{i},\vartheta }^{j}\rightarrow \chi _{\vartheta }^{j}\qquad \text {as } \Delta _{i}\rightarrow 0 \end{aligned}$$\end{document}

for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\vartheta }^{j}\in $$\end{document} Lip with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\chi _{\vartheta }^{j})'$$\end{document} bounded.

Let two characteristics \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\vartheta }^{1}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\vartheta }^{2}$$\end{document} intersect with the cone boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _{\vartheta }$$\end{document} and the leading shock \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\vartheta }$$\end{document} at points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t_{\varepsilon _{1}}^1,\chi _{\vartheta }^{1}(t_{\varepsilon _{1}}^1))$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t_{\varepsilon _{1}}^2,\chi _{\vartheta }^{2}(t_{\varepsilon _{1}}^2))$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\varepsilon _{1}}^1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\varepsilon _{1}}^2$$\end{document} , respectively. Then, as in [25], we apply the approximate conservation law to the domain below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\Delta _{i},\vartheta }^{1}$$\end{document} and above \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{\Delta _{i},\vartheta }^{1}$$\end{document} and use Lemma 7.1 to obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}_{j,\Delta _{i},\vartheta }(x-)\leqq C\sum _{h>[x_{\varepsilon _{1}}/\Delta x]} E_{\Delta _{i},\vartheta }(\Lambda _{h,n})<C\varepsilon _{1} \end{aligned}$$\end{document}

for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>t_{\varepsilon _{1}}^1+t_{\varepsilon _{1}}^2$$\end{document} . This completes the proof. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Theorem 7.1

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{b}_{\infty }:=\lim _{x\rightarrow \infty }p^{b}(x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_{\infty }:=\lim _{x\rightarrow \infty }s_{\vartheta }(x)$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b'_{\infty }=\lim _{x\rightarrow \infty }(b_{\vartheta })'_{+}(x)$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\lim _{x\rightarrow \infty }\sup \left\{ \big |U_{\vartheta }(x,y)-{\tilde{U}}(\sigma ;s_{\infty },G(s_{\infty }))\big |\,:\,\chi _{\vartheta }(x)<y<b_{\vartheta }(x)\right\} =0,\\&\dfrac{1}{2}\big |{\tilde{U}}(b_{\infty }';s_{\infty },G(s_{\infty }))\big |^2+\dfrac{\gamma (p^{b}_{\infty })^{\frac{\gamma -1}{\gamma }}}{\gamma -1}=\dfrac{1}{2}+\dfrac{\gamma p_{\infty }^{\frac{\gamma -1}{\gamma }}}{\gamma -1},\\&{\tilde{U}}(b_{\infty }';s_{\infty },G(s_{\infty }))\cdot (-b_{\infty }',1)=0. \end{aligned}$$\end{document}

Proof

For every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in [x_{k-1},x_{k})$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\big |U_{\vartheta }(x,y)-{\tilde{U}}(\sigma ;s_{\Delta _{i},\vartheta },G(s_{\Delta _{i},\vartheta }))\big | +\big |{\tilde{U}}(b_{\Delta _{i},\vartheta }';s_{\Delta _{i},\vartheta },G(s_{\Delta _{i},\vartheta }))\cdot (-b_{\Delta _{i},\vartheta }',1)\big |\\&\quad \quad +\Big |\dfrac{1}{2}\big |{\tilde{U}}(b_{\Delta _{i},\vartheta }';s_{\Delta _{i},\vartheta },G(s_{\Delta _{i},\vartheta }))\big |^2 +\dfrac{\gamma (p^{b}_{\Delta x,k})^{\frac{\gamma -1}{\gamma }}}{\gamma -1}-\dfrac{1}{2}-\dfrac{\gamma p_{\infty }^{\frac{\gamma -1}{\gamma }}}{\gamma -1}\Big |\\&\quad \leqq C \Big (\sum _{j=1}^{2}{\mathcal {L}}_{j,\Delta _{i},\vartheta }(x-)+|\Delta _{i}|\Big ). \end{aligned}$$\end{document}

By Theorem 6.1, letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\rightarrow \infty $$\end{document} , we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sup _{\chi _{\vartheta }(x)<y<b_{\vartheta }(x)} \big |U_{\vartheta }(x,\cdot )-{\tilde{U}}\big (\sigma ;s_{\vartheta },G(s_{\vartheta })\big )\big | +\big |{\tilde{U}}((b_{\vartheta })_{+}';s_{\vartheta },G(s_{\vartheta }))\cdot (-b_{\vartheta }',1)\big |\\&\quad \quad +\Big |\dfrac{1}{2}\big |{\tilde{U}}((b_{\vartheta })_{+}';s_{\vartheta },G(s_{\vartheta }))\big |^2+\dfrac{\gamma (p^{b})^{\frac{\gamma -1}{\gamma }}}{\gamma -1}-\dfrac{1}{2}-\dfrac{\gamma p_{\infty }^{\frac{\gamma -1}{\gamma }}}{\gamma -1}\Big |\\&\quad \leqq C \sum _{j=1}^{2}{\mathcal {L}}_{j,\vartheta }(x-). \end{aligned}$$\end{document}

Then, using Lemma 7.2 and noting that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{U}}(\sigma ;s,G(s))$$\end{document} is a continuous function with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} and s, we conclude our result. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}