Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

TL;DR

This paper proves the existence of a time periodic solution for the compressible Euler equations describing supersonic flow under a periodic external force, addressing a fundamental but underexplored problem in conservation laws.

Contribution

It establishes the existence of time periodic solutions for the compressible Euler equations with periodic outer forces, a novel result in the study of conservation laws.

Findings

Existence of time periodic solutions for the system.

Application to supersonic flow scenarios.

Addresses a gap in conservation law research.

Abstract

We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.

Equations

$$\{\begin{array}{l}{\rho }_t+m_x=0,\\ m_t+{\left(\frac{m^2}{\rho }+p(\rho )\right)}_x=F(x,t)\rho ,\end{array}x\in (0,1),t\in (0,1)$$

$$(\rho ,m){|}_{t=0}=({\rho }_0(x),m_0(x))(\rho ,m){|}_{x=0}=({\rho }_b,m_b).$$

$$\displaystyle\left\{\begin{array}[]{lll}u_{t}+f(u)_{x}=g(x,t,u),\quad{x}\in(0,1),\quad t\in(0,1),\\ u|_{t=0}=u_{0}(x),\\ u|_{x=0}=u_{b}\end{array}\right.$$

$$w:=\frac{m}{\rho }+\frac{{\rho }^{\theta }}{\theta }=v+\frac{{\rho }^{\theta }}{\theta },z:=\frac{m}{\rho }-\frac{{\rho }^{\theta }}{\theta }=v-\frac{{\rho }^{\theta }}{\theta }\left(\theta =\frac{\gamma -1}{2}\right).$$

$$|w|\geqq |z|,w\geqq 0,\text{mbox}\mathrm{when}v\geqq 0.|w|\leqq |z|,z\leqq 0,\text{mbox}\mathrm{when}v\leqq 0.$$

$$v=\frac{w+z}{2},\rho ={\left(\frac{\theta (w-z)}{2}\right)}^{1/\theta },m=\rho v.$$

$${\int }_0^1{\int }_0^1\rho ({\phi }_1{)}_t+m({\phi }_1{)}_{x}dxdt+{\int }_0^1{\rho }_0(x)\left({\phi }_1(x,0)-{\phi }_1(x,1)\right)dx$$

$$+{\int }_0^1{m}_b{\phi }_1(0,t)dt=0,$$

$${\int }_0^1{\int }_0^{1}m({\phi }_2{)}_t+\left(\frac{m^2}{\rho }+p(\rho )\right)({\phi }_2{)}_x+F(x,t)\rho {\phi }_{2}dxdt$$

$$+{\int }_0^1{m}_0(x)\left({\phi }_2(x,0)-{\phi }_2(x,1)\right)dx+{\int }_0^1\left\{\frac{{\left(m_b\right)}^2}{{\rho }_b}+p({\rho }_b)\right\}{\phi }_2(0,t)dt=0$$

$${\int }_0^1{\int }_0^1\eta (u){\psi }_t+q(u){\psi }_x+\nabla \eta (u)g(x,u)\psi dxdt\geqq 0$$

$$\frac{1}{\epsilon }{\int }_0^{\epsilon }f(u)dx\to f(u_b)\text{ as ε→0 in L∞(R+) weak∗}.$$

$$|F(x,t)|\leqq Kx\in [0,1].$$

$$M\geqq L\geqq \left(1+K\right).$$

$$0\leqq {\rho }_0(x),L-Kx\leqq z(u_0(x)),w(u_0(x))\leqq M+Kx,x\in (0,1)$$

$$L\leqq v_b-({\rho }_b{)}^{\theta }/\theta ,v_b+({\rho }_b{)}^{\theta }/\theta \leqq M,$$

$$\begin{array}{rl}& 0\leqq \rho (x,t),L-Kx\leqq z(u(x,t)),w(u(x,t))\leqq M+Kx,(x,t)\in (0,1)\times (0,1)\end{array}$$

$$z_t+{\lambda }_1{z}_x=F(x,t),w_t+{\lambda }_2{w}_x=F(x,t),$$

$${\lambda }_1=v-{\rho }^{\theta },{\lambda }_2=v+{\rho }^{\theta }.$$

$$\tilde{z}=z+Kx,\tilde{w}=w-Kx.$$

$${\tilde{z}}_t+{\lambda }_1{\tilde{z}}_x=F(x,t)+K{\lambda }_1,{\tilde{w}}_t+{\lambda }_2{\tilde{w}}_x=F(x,t)-K{\lambda }_2.$$

$$\begin{array}{r}L-Kx\leqq z,w\leqq M+Kx.\end{array}$$

$$L-Kx=z,w\leqq M+Kx$$

$${\lambda }_1=v-{\rho }^{\theta }\geqq v-\frac{{\rho }^{\theta }}{\theta }=z\geqq L-Kx.$$

$${\tilde{z}}_t+{\lambda }_1{\tilde{z}}_x=$$

$$\geqq$$

$$L-Kx\leqq z,w=M+Kx$$

$${\lambda }_2\geqq \frac{1+\theta }{2}M+\frac{1-\theta }{2}L+\theta Kx.$$

$${\tilde{w}}_t+{\lambda }_2{\tilde{w}}_x=$$

$$\leqq$$

$${\Delta }_x=\left\{(z,w);\rho \geqq 0,L-Kx\leqq z,w\leqq M+Kx\right\}$$

Full text

When we prove the existence of the time periodic solution, we face with two problems. One is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To overcome this, we employ the generalized invariant region, which depends on the space variables. This enable us to investigate the behavior of solutions in detail. Second is to construct a continuous map. We apply a fixed point theorem to the map from initial data to solutions after one period. Then, the map needs to be continuous. To construct this, we introduce the modified Lax-Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. The formula yields the desired map. Moreover, the invariant region grantees that it maps a compact convex set to itself. In virtue of the fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution. Finally, we apply the compensated compactness framework to prove the convergence of our approximate solutions.

N. Tsuge’s research is partially supported by Grant-in-Aid for Scientific Research (C) 17K05315, Japan.

1. Introduction

The present paper is concerned with the compressible Euler equation with an outer force.

[TABLE]

where $\rho$, $m$ and $p$ are the density, the momentum and the pressure of the gas, respectively. If $\rho>0$, $v=m/\rho$ represents the velocity of the gas. For a barotropic gas, $p(\rho)=\rho^{\gamma}/\gamma$, where $\gamma\in(1,5/3]$ is the adiabatic exponent for usual gases. The given function $F\in C^{1}([0,1]\times[0,1])$ represents a time periodic outer force with the time period $1$, i.e., $F(x,0)=F(x,1)$.

We consider the initial boundary value problem (1.1) with the initial and boundary data

[TABLE]

The above problem (1.1)–(1.2) can be written in the following form

[TABLE]

by using $u={}^{t}(\rho,m)$, $\displaystyle f(u)={}^{t}\!\left(m,\frac{m^{2}}{\rho}+p(\rho)\right)$ and $\displaystyle{g(x,u)={}^{t}\!\left(0,F(x,t)\rho\right)}$. We shall consider solutions with positive characteristic speeds. Therefore, from the Lopantinski condition, we do not supply a boundary condition at $x=1$.

In the present paper, we consider the compressible Euler equation. Let us survey the related mathematical results.

Concerning the one-dimensional initial value problem, DiPerna [3] proved the global existence by the vanishing viscosity method and a compensated compactness argument. The method of compensated compactness was introduced by Murat [6] and Tartar [9, 10]. DiPerna first applied the method to systems for the special case where $\gamma=1+2/n$ and $n$ is an odd integer. Subsequently, Ding, Chen and Luo [4] and Chen [1] extended his analysis to any $\gamma$ in $(1,5/3]$. In [5], Ding, Chen and Luo treated isentropic gas dynamics with a source term by using the fractional step procedure. By this result, the global existence theorem was obtained for the compressible Euler equation with an outer force. On the other hand, the stability (i.e., solutions are contained in a bounded set independent of the time variable) has not yet proved for a long time. Recently, in [15] and [18], it is proved by using the generalized invariant region. This method is also employed in [11]–[17].

In this paper, we consider a time periodic outer force and prove the existence of a time periodic solution. Although this problem is fundamental for other equations (ex. [7]), it has not been received much attention until now. For a single conservation law, Takeno [8] proved the existence of a time periodic solution for the space periodic boundary condition. However, unfortunately, a little is known for the system of conservation laws. This is reason why there are the following two problems. One is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To overcome this, we employ the generalized invariant region, which depends on the space variables. Second is to construct a continuous map in a finite dimension. We apply a fixed point theorem to a map from initial data to solutions after one period. Then, the map needs to be continuous. To construct this, we introduce the modified Lax-Friedrichs scheme, which has a recurrence formula (5.1) consisting of discretized approximate solutions. It yields the continuous map. In addition, the invariant region grantees that it maps a compact convex set to itself. Applying the Brouwer fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution.

To state our main theorem, we define the Riemann invariants $w,z$, which play important roles in this paper, as

Definition 1.1**.**

[TABLE]

These Riemann invariants satisfy the following.

Remark 1.1**.**

[TABLE]

From the above, the lower bound of $z$ and the upper bound of $w$ yield the bound of $\rho$ and $|v|$.

Moreover, we define the entropy weak solution.

Definition 1.2**.**

A measurable function $u(x,t)$ is called an time periodic entropy weak solution of the initial boundary value problem (1.6) with period $1$ if

[TABLE]

holds for any test function $\varphi_{1},\varphi_{2}\in C^{1}_{0}([0,1]\times[0,1])$ satisfying $\varphi_{1}(1,t)=\varphi_{2}(1,t)=0$ and

[TABLE]

holds for any non-negative test function $\psi\in C^{1}_{0}((0,1)\times(0,1))$, where $(\eta,q)$ is a pair of convex entropy–entropy flux of (1.1).

Remark 1.2**.**

It follows from the above definition that $f(u)$ converges in weak sense to $f(u_{b})$, i.e.,

[TABLE]

We choose a positive constant $K$ such that

[TABLE]

We then choose a positive value $L$ and $M$ such that

[TABLE]

Then our main theorem is as follows.

Theorem 1.1**.**

If $u_{0}=({\rho}_{0},{m}_{0})\in{L}^{\infty}(0,1)$ satisfy

[TABLE]

and the boundary data satisfy

[TABLE]

then the initial boundary value problem $(\ref{eqn:IP})$ has a time periodic entropy weak solution.

Moreover, the solution satisfies

[TABLE]

Remark 1.3**.**

If solutions satisfy (1.13), both of their characteristic speeds are positive.

1.1. Outline of the proof

The proof of main theorem is a little complicated. Therefore, before proceeding to the subject, let us grasp the point of the main estimate by a formal argument. We assume that a solution is smooth and the density is nonnegative in this section.

We consider the physical region $\rho\geqq 0$ (i.e., $w\geqq z$.). Recalling Remark 1.1, it suffices to derive the lower bound of $z(u)$ and the upper bound of $w(u)$ to obtain the bound of $u$. To do this, we diagonalize (1.1). If solutions are smooth, we deduce from (1.1)

[TABLE]

where $\lambda_{1}$ and $\lambda_{2}$ are the characteristic speeds defined as follows

[TABLE]

Moreover, set

[TABLE]

Then, it follows from (1.14) that

[TABLE]

Let us investigate the effects of the source term of $\eqref{eqn:force3}$ on sides AB and AC (see Fig. 1), where sides AB and AC represent $z=L-Kx$ and $w=M+Kx$ respectively. In these regions, $z$ and $w$ satisfy the following.

[TABLE]

First, we consider the source term of $\eqref{eqn:force3}_{1}$ on the side AB. Since the following

[TABLE]

hold on the side AB, we find that

[TABLE]

Therefore, we obtain

[TABLE]

Second, we consider the source term of $\eqref{eqn:force3}_{2}$ on the side AC. Since the following

[TABLE]

hold on the side AC, we find that

[TABLE]

Therefore, we obtain

[TABLE]

We thus conclude that the source term of $\eqref{eqn:force3}_{1}$ is positive on the side AB and the source term of $\eqref{eqn:force3}_{2}$ is negative on the side AC. We apply the maximum principle to $\tilde{z}$ and $\tilde{w}$. Then, if initial data are contained in the triangle ABC, we find that $\tilde{z}\geqq L$ and $\tilde{w}\leqq M$. It follows from that the solution remains in the same triangle. This implies that the following region

[TABLE]

is an invariant region for the initial boundary value problem $(\ref{eqn:IP})$. This idea is also used in [11]–[15].

Although the above argument is formal, it is essential. In fact, we shall implicitly use the property of the source terms in Section 4. However, we cannot justify the above argument by the standard difference scheme such as Godunov or Lax-Friedrichs scheme (cf. [4] and [5]). Therefore, we introduce the modified Lax Friedrichs scheme in Section 3.

The present paper is organized as follows. In Section 2, we review the Riemann problem and the properties of Riemann solutions. In Section 3, we construct approximate solutions by a modified Lax Friedrichs scheme. The approximate solutions consist of the steady state solutions and Riemann solutions stated in the previous section. In Section 4, we drive the bounded estimate of our approximate solutions. This section is the main point of the present paper. In Section 5, we apply a fixed point theorem and prove the existence of a fixed point.

2. Preliminary

In this section, we first review some results of the Riemann solutions for the homogeneous system of gas dynamics. Consider the homogeneous system

[TABLE]

A pair of functions $(\eta,q):{\bf R}^{2}\rightarrow{\bf R}^{2}$ is called an entropy–entropy flux pair if it satisfies an identity

[TABLE]

Furthermore, if, for any fixed ${m}/{\rho}\in(-\infty,\infty)$, $\eta$ vanishes on the vacuum $\rho=0$, then $\eta$ is called a weak entropy. For example, the mechanical energy–energy flux pair

[TABLE]

should be a strictly convex weak entropy–entropy flux pair.

The jump discontinuity in a weak solutions to (2.1) must satisfy the following Rankine–Hugoniot condition

[TABLE]

where $\lambda$ is the propagation speed of the discontinuity, $u_{0}=(\rho_{0},m_{0})$ and $u=(\rho,m)$ are the corresponding left and right state, respectively. A jump discontinuity is called a shock if it satisfies the entropy condition

[TABLE]

for any convex entropy pair $(\eta,q)$.

There are two distinct types of rarefaction and shock curves in the isentropic gases. Given a left state $(\rho_{0},m_{0})$ or $(\rho_{0},v_{0})$, the possible states $(\rho,m)$ or $(\rho,v)$ that can be connected to $(\rho_{0},m_{0})$ or $(\rho_{0},v_{0})$ on the right by a rarefaction or a shock curve form a 1-rarefaction wave curve $R_{1}(u_{0})$, a 2-rarefaction wave curve $R_{2}(u_{0})$, a 1-shock curve $S_{1}(u_{0})$ and a 2-shock curve $S_{2}(u_{0})$:

[TABLE]

respectively. Here we notice that shock wave curves are deduced from theRankine–Hugoniot condition (2.4).

Remark 2.1**.**

Assume that there exists $C>1$ such that

[TABLE]

Then, considering $w$ along $S_{1}(u_{0})$, we have

[TABLE]

where $O(1)$ depends only on $C$.

Considering $z$ along $S_{2}(u_{0})$, we similarly have

[TABLE]

where $O(1)$ depends only on $C$. These representation show that $S_{1}$ (resp. $S_{2}$) and $R_{1}$ (resp. $R_{2}$) have a tangency of second order at the point $(\rho_{0},u_{0})$.

2.1. Riemann Solution

Given a right state $(\rho_{0},m_{0})$ or $(\rho_{0},v_{0})$, the possible states $(\rho,m)$ or $(\rho,v)$ that can be connected to $(\rho_{0},m_{0})$ or $(\rho_{0},v_{0})$ on the left by a shock curve constitute 1-inverse shock curve $S_{1}^{-1}(u_{0})$ and 2-inverse shock curve $S_{2}^{-1}(u_{0})$:

[TABLE]

respectively.

Next we define a rarefaction shock. Given $u_{0},u$ on $S_{i}^{-1}(u_{0})\;(i=1,2)$, we call the piecewise constant solution to (2.1), which consists of the left and right states $u_{0},u$ a rarefaction shock. Here, notice the following: although the inverse shock curve has the same form as the shock curve, the underline expression in $S_{i}^{-1}(u_{0})$ is different from the corresponding part in $S_{i}(u_{0})$. Therefore the rarefaction shock does not satisfy the entropy condition.

We shall use a rarefaction shock in approximating a rarefaction wave. In particular, when we consider a rarefaction shock, we call the inverse shock curve connecting $u_{0}$ and $u$ a rarefaction shock curve.

From the properties of these curves in phase plane $(z,w)$, we can construct a unique solution for the Riemann problem

[TABLE]

where $x_{0}\in(-\infty,\infty)$, $\rho_{\pm}\geqq 0$ and $m_{\pm}$ are constants satisfying $|m_{\pm}|\leqq{C}\rho_{\pm}$. We denote the solution the Riemann solution $(u_{-},u_{+})$.

For the Riemann problem, the following invariant region exists.

Lemma 2.1**.**

For ${\rm B}_{+}\geqq{\rm B}_{-}$, the region $\sum({\rm B}_{+},{\rm B}_{-})=\{(\rho,\rho{v})\in{\bf R}^{2}:w=v+\rho^{\theta}/\theta,\>z=v-\rho^{\theta}/\theta,\>w\leqq{\rm B}_{+},\>z\geqq{\rm B}_{-},\>w-z\geqq 0\}$ is invariant with respect to both of the Riemann problem $(\ref{eqn:Riemann})$ and the average of the Riemann solutions in $x$. More precisely, if the Riemann data lie in $\sum({\rm B}_{+},{\rm B}_{-})$, the corresponding Riemann solutions $(\rho(x,t),m(x,t))=(\rho(x,t),\rho(x,t)v(x,t))$ lie in $\sum({\rm B}_{+},{\rm B}_{-})$, and their corresponding averages in $x$ are also in $\sum({\rm B}_{+},{\rm B}_{-})$, namely

[TABLE]

Lemma 2.3 can be found in [2, Lemma 3.3].

From the properties of these curves in phase plane $(z,w)$, we can construct a unique solution for the Riemann problem (2.1) and (2.7)

[TABLE]

and the Riemann initial boundary problem (2.1) and (2.8)

[TABLE]

and (2.1) and (2.9)

[TABLE]

where $x_{0}\in(0,1)$, $\rho_{\pm}\geqq 0$ and $m_{\pm}$ are constants satisfying $|m_{\pm}|\leqq{C}\rho_{\pm}$, $\lambda_{1}(u_{\pm})=v_{\pm}-(\rho_{\pm})^{\theta}\geqq 0$ and $\lambda_{1}(u_{b})=v_{b}-(\rho_{b})^{\theta}\geqq 0$.

For the problem (2.1) and (2.7), we can consult [2, Subsection 3.2].

Then the following theorem holds [2, Theorem 3.2].

Theorem 2.2**.**

There exists a unique piecewise smooth entropy solution $(\rho(x,t),$ $m(x,t))$ containing the vacuum state $(\rho=0)$ on the upper plane $t>0$ for each problem of (2.7), (2.8) and (2.9) satisfying

(1)* For the Riemann problem ((2.1)) and (2.7),*

[TABLE]

(2)* For the Riemann initial boundary problem ((2.1)) and (2.8),*

[TABLE]

(3)* For the Riemann initial boundary problem (2.9), the solution is $u_{-}$.*

Such solutions also have the following properties:

Lemma 2.3**.**

For ${\rm B}_{+}\geqq{\rm B}_{-}$, the region $\sum({\rm B}_{+},{\rm B}_{-})=\{(\rho,\rho{u})\in{\bf R}^{2}:w=u+\rho^{\theta}/\theta,\>z=u-\rho^{\theta}/\theta,\>w\leqq{\rm B}_{+},\>z\geqq{\rm B}_{-},\>w-z\geqq 0\}$ is invariant with respect to both of the Riemann problem (2.7), the Riemann initial boundary value problem (2.8) and the average of the Riemann solutions in $x$. More precisely, if the Riemann data lie in $\sum({\rm B}_{+},{\rm B}_{-})$, the corresponding Riemann solutions $(\rho(x,t),m(x,t))=(\rho(x,t),\rho(x,t)u(x,t))$ lie in $\sum({\rm B}_{+},{\rm B}_{-})$, and their corresponding averages in $x$ also in $\sum({\rm B}_{+},{\rm B}_{-})$, namely

[TABLE]

The proof of Lemma 2.2 can be found in [2, Lemma 3.3].

3. Construction of Approximate Solutions

In this section, we construct approximate solutions. In the strip $0\leqq{t}\leqq{1}$, we denote these approximate solutions by $u^{\varDelta}(x,t)=(\rho^{\varDelta}(x,t),m^{\varDelta}(x,t))$. For $N_{x}\in{\bf N}$, we define the space lengths by ${\varDelta}x=1/(2N_{x})$. Using $M$ in (1.10), we take time mesh length ${\varDelta}{x}$ such that

[TABLE]

where $[[x]]$ is the greatest integer not greater than $x$. Then we define $N_{t}=1/(2{\varDelta t})\in{\bf N}$. In addition, we set

[TABLE]

where ${\bf N}_{x}=\{0,1,2,\ldots,2N_{x}\}$ and ${\bf N}_{t}=\{0,1,2,\ldots,2N_{t}\}$.

First we define $u^{\varDelta}(x,-0)$ by

[TABLE]

and set

[TABLE]

Then, for $j\in J_{0}$, we define $E_{j}^{0}(u)$ by

[TABLE]

Next, assume that $u^{\varDelta}(x,t)$ is defined for $t<n{\varDelta}{t}$.

(i) $n$ is even

Then, for $j\in J_{n}$, we define $E^{n}_{j}(u)$ by

[TABLE]

(ii) $n$ is odd

Then, for $j\in J_{n}\setminus\{0,2N_{x}\}$, we define $E^{n}_{j}(u)$ by

[TABLE]

for $j\in\{0,2N_{x}\}$, we define $E^{n}_{j}(u)$ by

[TABLE]

Moreover, for $j\in J_{n}$, we define $u_{j}^{n}=(\rho_{j}^{n},m_{j}^{n})$ in the similar manner to (i).

Then, for $j\in J_{n}$, we define $u_{j}^{n}=(\rho_{j}^{n},m_{j}^{n})$ as follows.

We choose $\delta$ such that $1<\delta<1/(2\theta)$. If

[TABLE]

we define $u_{j}^{n}$ by $u_{j}^{n}=(0,0)$; otherwise, setting

[TABLE]

we define $u_{j}^{n}$ by

[TABLE]

Remark 3.1**.**

We find

[TABLE]

This implies that we cut off the parts where $z(E_{j}^{n}(u))<L-K({j{\varDelta}x})$ and $w(E_{j}^{n}(u))>M+K({j{\varDelta}x})$ in defining $z(u_{j}^{n})$ and ${w}(u_{j}^{n})$. Observing (4.1), the order of these cut parts is $o({\varDelta}x)$. The order is so small that we can deduce the compactness and convergence of our approximate solutions.

3.1. Construction of Approximate Solutions in the Cell

By using $u_{j}^{n}$ defined above, we construct the approximate solutions in the cell $n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}\quad(n\in{\bf N}_{t}),\quad(j-1){\varDelta}x\leqq{x}<(j+1){\varDelta}x\quad(j\in J_{n}\setminus\{0,1,2N_{x}-1,2N_{x}\})$.

We first solve a Riemann problem with initial data $(u_{j-1}^{n},u_{j+1}^{n})$. Call constants $u_{\rm L}(=u_{j-1}^{n}),u_{\rm M},u_{\rm R}(=u_{j+1}^{n})$ the left, middle and right states, respectively. Then the following four cases occur.

• •

Case 1 A 1-rarefaction wave and a 2-shock arise.

• •

Case 2 A 1-shock and a 2-rarefaction wave arise.

• •

Case 3 A 1-rarefaction wave and a 2-rarefaction arise.

• •

Case 4 A 1-shock and a 2-shock arise.

We then construct approximate solutions $u^{\varDelta}(x,t)$ by perturbing the above Riemann solutions. We consider only the case in which $u_{\rm M}$ is away from the vacuum. The other case (i.e., the case where $u_{\rm M}$ is near the vacuum) is a little technical. Therefore, we postpone the case near the vacuum to Appendix A. In addition, we omit the $L^{\infty}$ estimates for the case in this paper. The detail can be found in [11].

The case where $u_{\rm M}$ is away from the vacuum

Let $\alpha$ be a constant satisfying $1/2<\alpha<1$. Then we can choose a positive value $\beta$ small enough such that $\beta<\alpha$, $1/2+\beta/2<\alpha<1-2\beta$, $\beta<2/(\gamma+5)$ and $(9-3\gamma)\beta/2<\alpha$.

We first consider the case where $\rho_{\rm M}>({\varDelta}x)^{\beta}$, which means $u_{\rm M}$ is away from the vacuum. In this step, we consider Case 1 in particular. The constructions of Cases 2–4 are similar to that of Case 1.

Consider the case where a 1-rarefaction wave and a 2-shock arise as a Riemann solution with initial data $(u_{j}^{n},u_{j+1}^{n})$. Assume that $u_{\rm L},u_{\rm M}$ and $u_{\rm M},u_{\rm R}$ are connected by a 1-rarefaction and a 2-shock curve, respectively.

Step 1.

In order to approximate a 1-rarefaction wave by a piecewise constant rarefaction fan, we introduce the integer

[TABLE]

where $z_{\rm L}=z(u_{\rm L}),z_{\rm M}=z(u_{\rm M})$ and $[[x]]$ is the greatest integer not greater than $x$. Notice that

[TABLE]

Define

[TABLE]

and

[TABLE]

We next introduce the rays $x=(j+1/2){\varDelta}x+\lambda_{1}(z_{i}^{*},z_{i+1}^{*},w_{\rm L})(t-n{\varDelta}{t})$ separating finite constant states $(z_{i}^{*},w_{i}^{*})\;(i=1,\ldots,p)$, where

[TABLE]

[TABLE]

and

[TABLE]

We call this approximated 1-rarefaction wave a 1-rarefaction fan.

Step 2.

In this step, we replace the above constant states with the following functions of $x$:

Definition 3.1**.**

For given constants $x_{d}$ satisfying $(j-1){\varDelta}x\leqq{x_{d}}\leqq(j+1){\varDelta}x$ and

[TABLE]

satisfying $|m_{d}|\leqq{C}\rho_{d}$, we set

[TABLE]

Using $z(x)$ and $w(x)$, we define

[TABLE]

by the relation (1.8) as follows.

[TABLE]

We then define $\bar{\mathcal{U}}(x,x_{d},u_{d})$ with data $u_{d}$ at $x_{d}$ as (3.8) .

Moreover, for given functions $\bar{u}(x)$, we define $z(x,t)$ and $w(x,t)$ by

[TABLE]

Then, using $z(x,t)$ and $w(x,t)$, we define $u(x,t)=(\rho(x,t),m(x,t))$ in a similar manner to (3.9). We denote $u(x,t)$ by ${\mathcal{U}}(x,t;\bar{u})$.

Let $\bar{u}_{\rm L}(x)$ and $\bar{u}_{\rm R}(x)$ be $\bar{\mathcal{U}}(x,(j-1){\varDelta}x,u_{\rm L})$ and $\bar{\mathcal{U}}(x,(j+1){\varDelta}x,u_{\rm R})$, respectively. Set $\bar{u}_{1}(x):=\bar{u}_{\rm L}(x),\;u_{1}(x,t)={\mathcal{U}}(x,t;\bar{u}_{1}),\;u_{\rm R}(x,t)={\mathcal{U}}(x,t;\bar{u}_{\rm R})$ and $x_{1}:=(j-1){\varDelta}x.$

First, by the implicit function theorem, we determine a propagation speed $\sigma_{2}$ and $u_{2}=(\rho_{2},m_{2})$ such that

• (1.a)

$z_{2}:=z(u_{2})=z^{*}_{2}$

• (1.b)

the speed $\sigma_{2}$, the left state ${u}_{1}(x_{2},(n+1/2){\varDelta}t)$ and the right state $u_{2}$ satisfy the Rankine–Hugoniot conditions, i.e.,

[TABLE]

where $x_{2}:=j{\varDelta}x+\sigma_{2}{\varDelta}t/2$. Then we fill up by ${u}_{1}(x)$ the sector where $n{\varDelta}t\leqq{t}<(n+1){\varDelta}t,(j-1){\varDelta}x\leqq{x}<{j}{\varDelta}x+\sigma_{2}(t-n{\varDelta}t)$ (see Figure 3) and set $\bar{u}_{2}(x)=\bar{\mathcal{U}}(x,x_{2},u_{2})$ and $u_{2}(x,t)={\mathcal{U}}(x,t;\bar{u}_{2})$.

Assume that $u_{k}$, ${u}_{k}(x,t)$ and a propagation speed $\sigma_{k}$ with $\sigma_{k-1}<\sigma_{k}$ are defined. Then we similarly determine $\sigma_{k+1}$ and $u_{k+1}=(\rho_{k+1},m_{k+1})$ such that

• ($k$.a)

$z_{k+1}:=z(u_{k+1})=z^{*}_{k+1}$,

• ($k$.b)

$\sigma_{k}<\sigma_{k+1}$,

• ($k$.c)

the speed $\sigma_{k+1}$, the left state ${u}_{k}(x_{k+1},(n+1/2){\varDelta}t)$ and the right state $u_{k+1}$ satisfy the Rankine–Hugoniot conditions,

where $x_{k+1}:=j{\varDelta}x+\sigma_{k+1}{\varDelta}t/2$. Then we fill up by ${u}_{k}(x,t)$ the sector where $n{\varDelta}t\leqq{t}<(n+1){\varDelta}t,{j}{\varDelta}x+\sigma_{k}(t-{\varDelta}t)\leqq{x}<{j}{\varDelta}x+\sigma_{k+1}(t-n{\varDelta}t)$ (see Figure 3) and set $\bar{u}_{k+1}(x)=\bar{\mathcal{U}}(x,x_{k+1},u_{k+1})$ and $u_{k+1}(x,t)={\mathcal{U}}(x,t;\bar{u}_{k+1})$. By induction, we define $u_{i}$, ${u}_{i}(x,t)$ and $\sigma_{i}$ $(i=1,\ldots,p-1)$. Finally, we determine a propagation speed $\sigma_{p}$ and $u_{p}=(\rho_{p},m_{p})$ such that

• ($p$.a)

$z_{p}:=z(u_{p})=z^{*}_{p}$,

• ($p$.b)

the speed $\sigma_{p}$, and the left state ${u}_{p-1}(x_{p},(n+1/2){\varDelta}t)$ and the right state $u_{p}$ satisfy the Rankine–Hugoniot conditions,

where $x_{p}:=j{\varDelta}x+\sigma_{p}{\varDelta}t/2$. We then fill up by ${u}_{p-1}(x,t)$ and $u_{p}$ the sector where $n{\varDelta}t\leqq{t}<(n+1){\varDelta}t,{j}{\varDelta}x+\sigma_{p-1}(t-n{\varDelta}t)\leqq{x}<{j}{\varDelta}x+\sigma_{p}(t-n{\varDelta}t)$ and the line $n{\varDelta}t\leqq{t}<(n+1){\varDelta}t,x={j}{\varDelta}x+\sigma_{p}(t-n{\varDelta}t)$, respectively.

Given $u_{\rm L}$ and $z_{\rm M}$ with $z_{\rm L}\leqq{z}_{\rm M}$, we denote this piecewise functions of $x$ 1-rarefaction wave by $R_{1}^{\varDelta}(z_{\rm M})(u_{\rm L})$. Notice that from the construction $R^{\varDelta}_{1}(z_{\rm M})(u_{\rm L})$ connects $u_{\rm L}$ and $u_{p}$ with $z_{p}=z_{\rm M}$.

Now we fix ${u}_{\rm R}(x,t)$ and ${u}_{p-1}(x,t)$. Let $\sigma_{s}$ be the propagation speed of the 2-shock connecting $u_{\rm M}$ and $u_{\rm R}$. Choosing ${\sigma}^{\diamond}_{p}$ near to $\sigma_{p}$, ${\sigma}^{\diamond}_{s}$ near to $\sigma_{s}$ and $u^{\diamond}_{\rm M}$ near to $u_{\rm M}$, we fill up by ${u}^{\diamond}_{\rm M}(x,t)={\mathcal{U}}(x,t,\bar{u}^{\diamond}_{\rm M})$ the gap between $x=j{\varDelta}x+{\sigma}^{\diamond}_{p}(t-n{\varDelta}{t})$ and $x=j{\varDelta}x+{\sigma}^{\diamond}_{s}(t-n{\varDelta}{t})$, such that

• (M.a)

$\sigma_{p-1}<\sigma^{\diamond}_{p}<\sigma^{\diamond}_{s}$,

• (M.b)

the speed ${\sigma}^{\diamond}_{p}$, the left and right states ${u}_{p-1}(x^{\diamond}_{p},(n+1/2){\varDelta}t),{u}^{\diamond}_{\rm M}(x^{\diamond}_{p},(n+1/2){\varDelta}t)$ satisfy the Rankine–Hugoniot conditions,

• (M.c)

the speed ${\sigma}^{\diamond}_{s}$, the left and right states ${u}^{\diamond}_{\rm M}(x^{\diamond}_{s},(n+1/2){\varDelta}t),{u}_{\rm R}(x^{\diamond}_{s},(n+1/2){\varDelta}t)$ satisfy the Rankine–Hugoniot conditions,

where $\bar{u}^{\diamond}_{\rm M}(x)=\bar{\mathcal{U}}(x,j{\varDelta}x,u^{\diamond}_{\rm M})$, $x^{\diamond}_{p}:=j{\varDelta}x+\sigma^{\diamond}_{p}{\varDelta}/2$ and $x^{\diamond}_{s}:=j{\varDelta}x+\sigma^{\diamond}_{s}{\varDelta}/2$.

We denote this approximate Riemann solution, which consists of (3.8), by ${u}^{\varDelta}(x,t)$. The validity of the above construction is demonstrated in [11, Appendix A].

Remark 3.2**.**

${u}^{\varDelta}(x,t)$ satisfies the Rankine–Hugoniot conditions at the middle time of the cell, $t_{\rm M}:=(n+1/2){\varDelta}t$.

Remark 3.3**.**

The approximate solution $u^{\varDelta}(x,t)$ is piecewise smooth in each of the divided parts of the cell. Then, in the divided part, $u^{\varDelta}(x,t)$ satisfies

[TABLE]

3.2. Construction of Approximate Solutions near the Boundary

For $n\in{\bf N}_{t}$, separating four parts, we construct approximate solutions near the boundary as follows.

1. (i)

Near the left boundary $x=0$

1. (a)

$0\leqq{x}<2{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$, $n$ is even

We first solve a Riemann problem with initial data

[TABLE]

For the Riemann solution, we construct the approximate solutions in the cell $-2{\varDelta}x\leqq{x}<2{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ in the same manner to Section 3.1. Then we define the resultant approximate solution in $0\leqq{x}<2{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ by $u^{\varDelta}(x,t)$ in this area. 2. (b)

$0\leqq{x}<{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$, $n$ is odd

We first solve a Riemann problem with initial data

[TABLE]

For the Riemann solution, we construct the approximate solutions in the cell $-{\varDelta}x\leqq{x}<{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ in the same manner to Section 3.1. Then we define the resultant approximate solution in $0\leqq{x}<{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ by $u^{\varDelta}(x,t)$ in this area. 2. (ii)

Near the right boundary $x=1$

1. (a)

$1-2{\varDelta}x\leqq{x}<1,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$, $n$ is even

We first solve a Riemann problem with initial data

[TABLE]

For the Riemann solution, we construct the approximate solutions in the cell $1-2{\varDelta}x\leqq{x}<1+2{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ in the same manner to Section 3.1. Then we define the resultant approximate solution in $1-2{\varDelta}x\leqq{x}<1,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ by $u^{\varDelta}(x,t)$ in this area. 2. (b)

$1-{\varDelta}x\leqq{x}<1,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$, $n$ is odd

We first solve a Riemann problem with initial data

[TABLE]

For the Riemann solution, we construct the approximate solutions in the cell $1-{\varDelta}x\leqq{x}<1+{\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ in the same manner to Section 3.1. Then we define the resultant approximate solution in $1-{\varDelta}x\leqq{x}<1,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}$ by $u^{\varDelta}(x,t)$ in this area.

4. $L^{\infty}$ Estimate of the Approximate Solutions

We estimate Riemann invariants of $u^{\varDelta}(x,t)$ to use the invariant region theory. Our aim in this section is to deduce from (3.2) the following theorem:

Theorem 4.1**.**

[TABLE]

where ${\it o}({\varDelta}x)$ depends only on $M$.

In this section, we first assume

[TABLE]

instead of (1.10), where $\varepsilon$ is any fixed positive value.

Throughout this paper, by the Landau symbols such as $O({\varDelta}x)$, $O(({\varDelta}x)^{2})$ and $o({\varDelta}x)$, we denote quantities whose moduli satisfy a uniform bound depending only on $M$ and $X$ unless we specify otherwise. In addition, for simplicity, we denote $w(\bar{u}_{i}(x))$ and $z(\bar{u}_{i}(x))$ by $\bar{w}_{i}(x)$ and $\bar{z}_{i}(x)$.

Now, in the previous section, we have constructed $u^{\varDelta}(x,t)$. Then, the following four cases occur.

• •

In Case 1, the main difficulty is to obtain $(\ref{eqn:goal})_{1}$ along $R^{\varDelta}_{1}$.

• •

In Case 2, the main difficulty is to obtain $(\ref{eqn:goal})_{2}$ along $R^{\varDelta}_{2}$.

• •

In Case 3, (4.1) follows that of Case 1 and Case 2.

• •

In Case 4, (4.1) is easier than that of the other cases.

Thus we treat Case 1 in particular. In Case 1, we derive $(\ref{eqn:goal})_{1}$ along $R^{\varDelta}_{1}$ and estimate the other quasi-steady state solutions. We can estimate the other cases in a fashion similar to Case 1.

4.1. Estimates of $\bar{u}^{\varDelta}(x,t)$ in Case 1

In this step, we estimate ${u}^{\varDelta}(x,t)$ in Subsection 3.1. In this case, each component of $\bar{u}^{\varDelta}(x,t)$ has the following properties, which is proved in [11, Appendix A]:

[TABLE]

Now we derive (4.1) in the interior cell $n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t},\;(j-1){\varDelta}x\leqq{x}\leqq(j+1){\varDelta}x$. To do this, we first consider components of $\bar{u}^{\varDelta}(x,t)$.

Estimate of $\bar{z}_{i}(x)\;(i=1,\ldots,p-1)$.

Recalling that $z_{1}=z_{\rm L}=z_{j-1}^{n}\geqq L-K({(j-1){\varDelta}x})$, we have

[TABLE]

On the other hand, the construction of $\bar{u}_{i}(x)$ implies that $z_{i}=z_{1}+(i-1)({\varDelta}x)^{\alpha}$. If $i\geqq 2$, since $\alpha<1$, it follows that

[TABLE]

We thus obtain

[TABLE]

Estimate of $\bar{z}_{\rm R}(x)$.

Recall that $z_{\rm R}=z_{j+1}^{n}\geqq L-K((j+1){\varDelta}x)$. We then have

[TABLE]

Estimate of $\bar{z}^{\diamond}_{\rm M}(x)$.

If ${u}^{\diamond}_{\rm M}(x^{\diamond}_{s},(n+1/2){\varDelta}t)$ and ${u}_{\rm R}(x^{\diamond}_{s},(n+1/2){\varDelta}t)$ are connected by a 2-shock curve, we find

[TABLE]

Then, in view of (3.10), we obtain

[TABLE]

Therefore, from (4.10), we have

[TABLE]

On the other hand, we consider the case where ${u}^{\diamond}_{\rm M}(x^{\diamond}_{s},(n+1/2){\varDelta}t)$ and ${u}_{\rm R}(x^{\diamond}_{s},(n+1/2){\varDelta}t)$ are connected by a 2-rarefaction shock curve. First, recall that $u_{\rm M}$ and $u_{\rm R}$ are connected, not by a 2-rarefaction shock curve but by a 2-shock curve. Since $|{u}^{\diamond}_{\rm M}(x^{\diamond}_{s},(n+1/2){\varDelta}t)-\bar{u}^{\diamond}_{\rm M}(x^{\diamond}_{s})|=O({\varDelta}x)$, $|\bar{u}^{\diamond}_{\rm M}(x^{\diamond}_{s})-u^{\diamond}_{\rm M}|=O({\varDelta}x)$, $|{u}_{\rm R}(x^{\diamond}_{s},(n+1/2){\varDelta}t)-\bar{u}_{\rm R}(x^{\diamond}_{s})|=O({\varDelta}x)$ and $|\bar{u}_{\rm R}(x^{\diamond}_{s})-u_{\rm R}|=O({\varDelta}x)$, we then deduce from (4.7) that

[TABLE]

Therefore, from Remark 2.1 and the fact that $\beta<2/(\gamma+5)$, we conclude that

[TABLE]

Therefore, from (4.11)–(4.12), we obtain

[TABLE]

Estimate of $\bar{w}_{i}(x)\;(i=1,\ldots,p-1)$.

First, we recall that

[TABLE]

We then assume that

[TABLE]

It follows that

[TABLE]

From (4.6), we obtain

[TABLE]

Therefore, (4.14) holds for any $i$.

In view of (3.3) and (4.14), since $3\alpha-(\gamma-1)\beta>1$, we thus conclude that

[TABLE]

Estimate of $\bar{w}^{\diamond}_{\rm M}(x)$.

Combining the fact that $(9-3\gamma)\beta/2<\alpha$, (4.8) and (4.16), we thus have

[TABLE]

Estimate of $\bar{w}_{\rm R}(x)$.

Recalling $w_{\rm R}=w(u^{n}_{j+1})\leqq M+K((j+1){\varDelta}x)$, it follows that

[TABLE]

Estimate of $u^{\varDelta}(x,t)$ in the interior cell $n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t},\;(j-1){\varDelta}x\leqq{x}<(j+1){\varDelta}x$.

Let us derive $\eqref{eqn:goal}_{1}$.

Estimate 1

We first consider the case where $z(\bar{u}^{\varDelta}(x,t))\geqq L-K{x}+\sqrt{{\varDelta}x}$. On the other hand, we recall that $z({u}^{\varDelta}(x,t))=z(\bar{u}^{\varDelta}(x,t))+{\it O}({\varDelta}x)$. Therefore, choosing ${\varDelta}x$ small enough, we conclude $\eqref{eqn:goal}_{1}$.

Estimate 2

We next consider the case where

[TABLE]

This case is the validity of Subsection 1.1. Recalling its argument, let us deduce $\eqref{eqn:goal}_{1}$.

From (4.9), (4.10), (4.13), (4.16), (4.17) and (4.18), we have

[TABLE]

Then, from (4.19)–(4.20), choosing ${\varDelta}x$ small enough, we find that

[TABLE]

Then, from (1.9) and (4.2), we obtain

[TABLE]

As a result, from (4.20), we drive $\eqref{eqn:goal}_{1}$. We can similarly obtain $\eqref{eqn:goal}_{2}$.

5. Recurrence formula

From Remark 3.3, $u^{\varDelta}$ satisfy

[TABLE]

on the divided part in the cell where $u^{\varDelta}$ are smooth. Moreover, $u^{\varDelta}$ satisfy an entropy condition (see [11, Lemma 5.1–Lemma 5.4]) along discontinuous lines approximately. Then, applying the Green formula to $(u^{\varDelta})_{t}+f(u^{\varDelta})_{x}-g(x,t,u^{\varDelta})$ in the cell $(j-1){\varDelta}x\leqq{x}<(j+1){\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}\quad(j\in J_{n+1},\;n\in{\bf N}_{t})$, we have

[TABLE]

where $t_{n}=n{\varDelta t}$ and

[TABLE]

Moreover, from (4.1) and [12, Lemma 6.1], we have

[TABLE]

Then, we define a sequence $\tilde{u}^{n}_{j}=(\tilde{\rho}^{n}_{j},\tilde{m}^{n}_{j})$ as follows.

[TABLE]

Therefore, from (5.1)–(5.3), there exists $\delta({\varDelta}x)>0$ satisfying $\delta({\varDelta}x)\rightarrow 0$ as ${\varDelta}x\rightarrow 0$, such that

[TABLE]

Then, we define a map $F:{\bf R}^{4N_{x}+2}\rightarrow{\bf R}^{4N_{x}+2}$ as follows.

[TABLE]

From (5.3), $F$ is continuous. Moreover, from (5.5), $F$ is the map from a bounded set to the same bounded set. Therefore, applying the Brouwer fixed point theorem to $F$, we have a fixed point $(\tilde{u}^{0}_{j})^{*}$. We supply $(\tilde{u}^{0}_{j})^{*}$ as initial data ${u}^{0}_{j}$ for our approximate solutions.

The following proposition and theorem can be proved in the same manner to [11]–[13].

Proposition 5.1**.**

The measure sequence

[TABLE]

lies in a compact subset of $H_{\rm loc}^{-1}(\Omega)$ for all weak entropy pair $(\eta,q)$, where $\Omega\subset{\bf R}\times{\bf R}_{+}$ is any bounded and open set.

Theorem 5.2**.**

Assume that the approximate solutions $(\rho^{\varDelta},m^{\varDelta})$ satisfy Theorem 4.1 and Proposition 5.1. Then there is a convergent subsequence in the approximate solutions $(\rho^{\varDelta}(x,t),m^{\varDelta}(x,t))$ such that

[TABLE]

The function $(\rho(x,t),m(x,t))$ is a time periodic entropy solution of the initial boundary value problem (1.6).

We have proved Theorem 1.1 under the condition (4.2). However, since $\varepsilon$ are arbitrary, we conclude Theorem 4.1 under the condition (1.10).

6. Open problem for a time-periodic outer force

In this section, we introduce some open problem related to (1.11). The most difficult point of the present problems is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To solve this problem, we have introduced the generalized invariant region depending on the space variable. Observing (1.13), we find that the upper bound $U(x)$ and lower bound $L(x)$ are monotone functions (In (1.13), $U(x)=M+Kx$ and $L(x)=L-Kx$ in particular.). This method is used in [11]–[18] and these bounds are also monotone functions. The method is useful for the Cauchy problem or the initial boundary problem for a half space. Since we need not supply the boundary condition at $x=1$ for the present paper, our problem is the almost same as the initial boundary value problem for a half space. We can thus construct the invariant region.

However, unfortunately, our method is not useful for the periodic boundary condition or the initial boundary value problem for a bounded interval. Considering the periodic boundary condition, $U$ and $L$ need to be periodic. Next, considering the boundary condition $m|_{x=0}=m|_{x=1}=0$ instead of (1.12), $U(0)\geq-L(0)$ and $U(1)\leq-L(1)$ need to hold. Recalling our upper and lower bound are monotone, $U(x)$ and $L(x)$ cannot satisfy these conditions. On the other hand, as mentioned above, we can construct invariant regions for the Cauchy problem and the initial boundary value problem for a half space. However, since their regions are not finite, we can not apply the Brouwer fixed point theorem like (5.6). Therefore, the existence of a time periodic solution for these problems is still open.

Appendix A Construction of Approximate Solutions near the vacuum

In this step, we consider the case where $\rho_{\rm M}\leqq({\varDelta}x)^{\beta}$, which means that $u_{\rm M}$ is near the vacuum. In this case, we cannot construct approximate solutions in a similar fashion to Subsection 3.1. Therefore, we must define $u^{\varDelta}(x,t)$ in a different way.

In this appendix, we define our approximate solutions in the cell $(j-1){\varDelta}x\leqq{x}<(j+1){\varDelta}x,\;n{\varDelta}{t}\leqq{t}<(n+1){\varDelta}{t}\quad(j\in{\bf Z},\>n\in{\bf Z}_{\geqq 0})$. We set $L_{j}:=L-K((j+1){\varDelta}x)$ and $U_{j}:={M}+K((j-1){\varDelta}x)$.

Case 1 A 1-rarefaction wave and a 2-shock arise.

In this case, we notice that $\rho_{\rm R}\leqq({\varDelta}x)^{\beta},\;z_{\rm R}\geqq L_{j}$ and $w_{\rm R}\leqq U_{j}$.

Definition of $\bar{u}^{\varDelta}$ in Case 1

Case 1.1 $\rho_{\rm L}>({\varDelta}x)^{\beta}$

We denote $u^{(1)}_{\rm L}$ a state satisfying $w(u_{\rm L}^{(1)})=w(u_{\rm L})$ and $\rho^{(1)}_{\rm L}=({\varDelta}x)^{\beta}$. Let $u^{(2)}_{\rm L}$ be a state connected to $u_{\rm L}$ on the right by $R_{1}^{\varDelta}(z^{(1)}_{\rm L})(u_{\rm L})$. We set

[TABLE]

Then, we define

[TABLE]

Case 1.2 $\rho_{\rm L}\leqq({\varDelta}x)^{\beta}$

(i) $z(u_{\rm L})\geqq{L}_{j}$

In this case, we define $u^{\varDelta}(x,t)$ as a Riemann solution $(u_{\rm L},u_{\rm R})$.

(ii) $z(u_{\rm L})<L_{j}$

In this case, recalling $z(u_{\rm L})=z(u^{n}_{j})\geqq L-K((j-1){\varDelta}x)$, we can choose $x^{(4)}$ such that $(j-1){\varDelta}x\leqq{x}^{(4)}\leqq(j+1){\varDelta}x$ and $z(u_{\rm L})-K({x^{(4)}}-{x_{\rm L}})=L_{j},$ where $x_{\rm L}:=(j-1){\varDelta}x$. We set

[TABLE]

In the region where $(j-1){\varDelta}x\leqq{x}\leqq{j}{\varDelta}x+\lambda_{1}(u^{(4)}_{\rm L})(t-n{\varDelta}t)$ and $n{\varDelta}t\leqq{t}<(n+1){\varDelta}t$, we define $\bar{u}^{\varDelta}(x,t)$ as a solution of (3.8) such that $\bar{u}^{\varDelta}((j-1){\varDelta}x)=u_{\rm L}$. We next solve a Riemann problem $(u^{(4)}_{\rm L},u_{\rm R})$. In the region where ${j}{\varDelta}x+\lambda_{1}(u^{(4)}_{\rm L})(t-n{\varDelta}t)\leqq{x}\leqq(j+1){\varDelta}x$ and $n{\varDelta}t\leqq{t}<(n+1){\varDelta}t$, we define $\bar{u}^{\varDelta}(x,t)$ as this Riemann solution.

Definition of ${u}^{\varDelta}$

Case 1 In the region where $\bar{u}^{\varDelta}(x,t)$ is the Riemann solution, we define $u^{\varDelta}(x,t)$ by $u^{\varDelta}(x,t)=\bar{u}^{\varDelta}(x,t)$; otherwise, the definition of $u^{\varDelta}(x,t)$ is similar to Subsection 3.1. Thus, for a Riemann solution near the vacuum, we define our approximate solution as the Riemann solution itself.

Case 2 A 1-shock and a 2-rarefaction wave arise.

From symmetry, this case reduces to Case 1.

Case 3 A 1-rarefaction wave and a 2-rarefaction wave arise.

For $u_{\rm L}$ of Case 1, we define $u^{*}_{\rm L}$ and $\lambda^{*}_{\rm L}$ as follows.

[TABLE]

where $\lambda_{1}(u)$ be the 1-characteristic speed of $u$. Then, for $u_{\rm L}$ of Case 3, we can determine $u^{*}_{\rm L}$ and $\lambda^{*}_{\rm L}$ in a similar manner to Case 1. From symmetry, for $u_{\rm R}$ of Case 3, we can also determine $u^{*}_{\rm R}$ and $\lambda^{*}_{\rm R}$.

In the region $(j-1){\varDelta}x\leqq{x}\leqq{j}{\varDelta}x+\lambda^{*}_{\rm L}(t-n{\varDelta}t),\;{j}{\varDelta}x+\lambda^{*}_{\rm R}(t-n{\varDelta}t)\leqq{x}\leqq(j+1){\varDelta}x$ and $n{\varDelta}t\leqq{t}<(n+1){\varDelta}t$, we define $\bar{u}^{\varDelta}$ in a similar manner to Subsection 3.1. In the other region, we define $\bar{u}^{\varDelta}$ as the Riemann solution $(u^{*}_{\rm L},u^{*}_{\rm R})$.

We define ${u}^{\varDelta}$ in the same way as Case 1.

Case 4 A 1-shock and a 2-shock arise.

We notice that $z_{\rm L}\geqq L_{j},\;w_{\rm L}\leqq U_{j},\;z_{\rm R}\geqq L_{j}$ and $w_{\rm R}\leqq U_{j}$. In this case, we define $u^{\varDelta}(x,t)$ as the Riemann solution $(u_{\rm L},u_{\rm R})$.

We complete the construction of our approximate solutions.

Acknowledgements

The author would appreciate Prof. Shigeharu Takeno for his useful comments.