Radial transonic shock solutions of Euler-Poisson system in convergent nozzles

TL;DR

This paper proves the existence and uniqueness of a transonic shock solution for the Euler-Poisson system in a convergent nozzle under specific boundary conditions and electric field strength, advancing understanding of shock phenomena in electrohydrodynamics.

Contribution

It establishes the first rigorous proof of a unique transonic shock solution for the Euler-Poisson system in a convergent nozzle with specified boundary data.

Findings

Existence of a transonic shock solution under large electric field conditions.

Uniqueness of the shock solution in the given setting.

Dependence of solution properties on electric field strength and nozzle length.

Abstract

Given constant data of density $\rho_0$, velocity $-u_0{\bf e}_r$, pressure $p_0$ and electric force $-E_0{\bf e}_r$ for supersonic flow at the entrance, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit, we prove that Euler-Poisson system admits a unique transonic shock solution in a two dimensional convergent nozzle, provided that $u_0>0$, $E_0>0$, and that $E_0$ is sufficiently large depending on $(\rho_0, u_0, p_0)$ and the length of the nozzle.

Equations

$$\left\{\begin{array}[c]{l}\text{div}(\rho\mathbf{u})=0,\\ \text{div}(\rho\mathbf{u}\otimes\mathbf{u})+\nabla p=\rho\nabla\Phi,\\ \text{div}(\rho\mathcal{E}{\bf u}+p{\bf u})=\rho\mathbf{u}\cdot\nabla\Phi,\\ \Delta\Phi=\rho-\tilde{b},\end{array}\right.$$

$$p(\rho ,S)=e^S{\rho }^{\gamma }\text{and}\mathcal{E}(\mathbf{u},\rho ,S)=\frac{1}{2}|\mathbf{u}{|}^2+\frac{e^S{\rho }^{\gamma -1}}{\gamma -1},$$

$$[\rho \mathbf{u}\cdot {\mathbf{\nu }}_s{]}_{{\Gamma }_s}=[\mathbf{u}\cdot {\mathbf{\tau }}_s{]}_{{\Gamma }_s}=[\rho (\mathbf{u}\cdot {\mathbf{\nu }}_s{)}^2+p{]}_{{\Gamma }_s}=[B{]}_{{\Gamma }_s}=0,$$

$$[\Phi {]}_{{\Gamma }_s}=[\nabla \Phi \cdot {\mathbf{\nu }}_s{]}_{{\Gamma }_s}=0,$$

$$B=\frac{1}{2}|\mathbf{u}{|}^2+\frac{\gamma p}{(\gamma -1)\rho }=\frac{1}{2}|\mathbf{u}{|}^2+\frac{\gamma e^S{\rho }^{\gamma -1}}{\gamma -1}.$$

$$\begin{array}{rl}& 0<{\mathbf{u}}_{+}\cdot {\mathbf{\nu }}_s<{\mathbf{u}}_{-}\cdot {\mathbf{\nu }}_s\text{on}{\Gamma }_s,\\ \text{or equivalently}& {\mathbf{u}}_{+}\cdot {\mathbf{\nu }}_s>0,{\mathbf{u}}_{-}\cdot {\mathbf{\nu }}_s>0\text{and}{S}_{+}>S_{-}\text{on}{\Gamma }_s\end{array}$$

$$\{\begin{array}{l}\text{div}(\rho \mathbf{u})=0,\\ \text{div}(\rho \mathbf{u}\otimes \mathbf{u})+\nabla p=0,\\ \text{div}(\rho \mathcal{E}\mathbf{u}+p\mathbf{u})=0\end{array}$$

$$\mathcal{A}:=\{\mathrm{x}\in {\mathbb{R}}^2:r_1<|\mathrm{x}|<r_0\}.$$

$${\Gamma }_0:=\partial \mathcal{A}\cap \{|\mathrm{x}|=r_0\},{\Gamma }_1:=\partial \mathcal{A}\cap \{|\mathrm{x}|=r_1\}.$$

$$\Vert \tilde{b}{\Vert }_{C^1(\overline{\mathcal{A}})}\le b_0$$

$$(\rho ,\mathbf{u},p,\nabla \Phi )=({\rho }_0,-u_0{\mathbf{e}}_r,p_0,-E_0{\mathbf{e}}_r)\text{on}{\Gamma }_0$$

$$(\rho ,\mathbf{u},p,\nabla \Phi )(\mathrm{x})=(\tilde{\rho }(r),\tilde{u}(r){\mathbf{e}}_r,\tilde{p}(r),\tilde{E}(r){\mathbf{e}}_r),$$

$$\begin{array}{rl}& \{\begin{array}{l}\frac{d}{dr}(r\tilde{\rho }\tilde{u})=0\\ \frac{d}{dr}(r\tilde{\rho }{\tilde{u}}^2)+r\frac{d\tilde{p}}{dr}=r\tilde{\rho }\tilde{E}\\ \frac{d}{dr}(r\tilde{\rho }\tilde{u}B)=r\tilde{\rho }\tilde{u}\tilde{E}\\ \frac{d}{dr}(r\tilde{E})=r(\tilde{\rho }-\tilde{b}(r))\end{array}\text{for}{r}_1<r<r_0,\\ & (\tilde{\rho },\tilde{u},\tilde{p},\tilde{E})(r_0)=({\rho }_0,-u_0,p_0,-E_0).\end{array}$$

$$(t,\hat{t})=(r_0-r,r),$$

$$(\rho ,u,p,E,b)(t):=(\tilde{\rho },-\tilde{u},\tilde{p},-\tilde{E},\tilde{b})(r).$$

$$\begin{array}{rl}& \{\begin{array}{l}(\hat{t}\rho u{)}^{\prime }=0\\ (\hat{t}\rho u^2{)}^{\prime }+\hat{t}{p}^{\prime }=\hat{t}\rho E\\ (\hat{t}\rho uB{)}^{\prime }=\hat{t}\rho uE\\ (\hat{t}E{)}^{\prime }=\hat{t}(\rho -b)\end{array}\text{for}0<t<r_0-r_1(:=T),\\ & (\rho ,u,p,E)(0)=({\rho }_0,u_0,p_0,E_0),\end{array}$$

$$B=\frac{1}{2}{u}^2+\frac{\gamma p}{(\gamma -1)\rho },$$

$$[\rho u{]}_{{\Gamma }_s}=[\rho u^2+p{]}_{{\Gamma }_s}=[B{]}_{{\Gamma }_s}=[E{]}_{{\Gamma }_s}=0,$$

$${\Gamma }_s=\{t=t_s\}\text{for some ts∈(0,T).}$$

$$S^{\prime }=0.$$

$$(\hat{t}\rho u,e^S)=(m_0,{\kappa }_0)\text{on}[0,T]\text{for}(m_0,{\kappa }_0)=(r_0{\rho }_0{u}_0,\frac{p_0}{{\rho }_0^{\gamma }}).$$

$$c^2=\gamma {\kappa }_0{\rho }^{\gamma -1}=\gamma {\kappa }_0{\left(\frac{m_0^2}{\gamma {\kappa }_0}\right)}^{\frac{\gamma -1}{\gamma +1}}{\left(\frac{1}{{\hat{t}}^2{M}^2}\right)}^{\frac{\gamma -1}{\gamma +1}}=:\frac{1}{{\mu }_0}{\left(\frac{1}{{\hat{t}}^2{M}^2}\right)}^{\frac{\gamma -1}{\gamma +1}}.$$

$$B^{\prime }=E.$$

$$(M^2{)}^{\prime }=\frac{M^2}{(M^2-1)}\left((\gamma +1){\mu }_{0}E({\hat{t}}^2{M}^2{)}^{\frac{\gamma -1}{\gamma +1}}-\frac{1}{\hat{t}}(2+(\gamma -1)M^2)\right)=:h_1(t,M,E,{\kappa }_0).$$

$$\rho ={\left(\frac{1}{\gamma {\kappa }_0{\mu }_0}\right)}^{\frac{1}{\gamma -1}}{\left(\frac{1}{{\hat{t}}^2{M}^2}\right)}^{\frac{1}{\gamma +1}}=:{\mu }_1{\left(\frac{1}{{\hat{t}}^2{M}^2}\right)}^{\frac{1}{\gamma +1}},$$

$$(\hat{t}E{)}^{\prime }=\hat{t}\left({\mu }_1{\left(\frac{1}{{\hat{t}}^2{M}^2}\right)}^{\frac{1}{\gamma +1}}-b(t)\right)=:h_2(t,M,E,{\kappa }_0).$$

$$\{\begin{array}{l}(\hat{t}\rho u,e^S)=(m_0,{\kappa }_0),\\ (M^2{)}^{\prime }=h_1(t,M,E,{\kappa }_0),\\ (\hat{t}E{)}^{\prime }=h_2(t,M,E,{\kappa }_0),\end{array}$$

$$\{\begin{array}{l}(\hat{t}\rho u,e^S)=(m_0,{\kappa }_0),\\ {\rho }^{\prime }=g_1(t,\rho ,E,{\kappa }_0),\\ (\hat{t}E{)}^{\prime }=g_2(t,\rho ,E),\end{array}$$

$$(g_1,g_2)(t,\rho ,E,{\kappa }_0)=\left(\frac{\rho (\hat{t}E-\frac{m_0^2}{{\hat{t}}^2{\rho }^2})}{\hat{t}(\gamma {\kappa }_0{\rho }^{\gamma -1}-\frac{m_0^2}{{\hat{t}}^2{\rho }^2})},\hat{t}(\rho -b(t))\right).$$

$${\rho }_{-}>0,u_{-}>0,\text{and}{M}_{-}=\frac{u_{-}}{\sqrt{\gamma {\kappa }_0{\rho }_{-}^{\gamma -1}}}>1\text{for}0\le t\le T.$$

$$(\rho ,u,p,E)(t;t_s)=\{\begin{array}{l}({\rho }_{-},u_{-},p_{-},E_{-})(t)\text{for 0≤t<ts}\\ ({\rho }_{+},u_{+},p_{+},E_{+})(t;t_s)\text{for ts≤t≤T}\end{array},$$

Full text

The first author is supported in part by Samsung Science and Technology Foundation under Project Number SSTF-BA1502-02. The second author supported in part under the framework of international cooperation program managed by National Research Foundation of Korea(NRF-2015K2A2A2002145).

∗ Corresponding author: xxxx

Myoungjean Bae∗

Department of Mathematics, POSTECH, Pohang, Gyungbuk, 37673, Republic of Korea,

Korea Institute for Advanced Study 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea

Yong Park

Department of Mathematics, POSTECH, Pohang, Gyungbuk, 37673, Republic of Korea,

(Communicated by )

1. Introduction

The steady Euler-Poisson system

[TABLE]

describes the motion of electrons governed by self-generated electric field in macroscopic scale. In (1.1), ${\bf u},\rho$, $p$, and $\mathcal{E}$ represent the macroscopic particle velocity, density, pressure, and the total energy density, respectively. And, $\Phi$ represents the electric potential generated by the Coulomb force of particles. The function $\tilde{b}>0$ is fixed, and represents the density of positively charged background ions. In this work, we consider ideal polytropic gas for which the pressure $p$ and the energy density $\mathcal{E}$ are given by

[TABLE]

respectively. Here, the constant $\gamma>1$ is called the adiabatic exponent, and $S>0$ represents the entropy.

The goal of this work is to construct a family of radial transonic shock solutions to (1.1) in a two dimensional convergent nozzle, and to study various analytical features especially including the monotonicity property of the pressure at the exit with respect to shock location, provided that the magnitude of electric field at the entrance is fixed sufficiently large.

Definition 1.1** (A shock solution of E-P system).**

Let $\Omega\subset\mathbb{R}^{2}$ be an open and connected set, and suppose that a $C^{1}$ curve $\Gamma_{s}$ divides $\Omega$ into two open and connected sub-domains $\Omega^{+}$ and $\Omega^{-}$ so that $\Omega^{+}\cup\Gamma_{s}\cup\Omega^{-}=\Omega$. Let ${{\bm{\nu}}_{s}}$ be the unit normal vector field on $\Gamma_{s}$ oriented into $\Omega^{+}$, and let ${\bm{\tau}}_{s}$ be a tangent vector field on $\Gamma_{s}$. We call ${\bf U}:=(\rho,{\bf u},p,\Phi)\in[L^{\infty}_{\rm{loc}}(\Omega)\cap C^{0}(\overline{\Omega^{\pm}})\cap C^{1}(\Omega^{\pm})]^{4}\times[W^{1,\infty}_{\rm{loc}}(\Omega)\cap C^{1}(\overline{\Omega^{\pm}})\cap C^{2}(\Omega^{\pm})]$ with $\rho>0$ in $\overline{\Omega}$ a shock solution to (1.1) in $\Omega$ with a shock $\Gamma_{s}$ if ${\bf U}$ satisfies (1.1) pointwisely in $\Omega^{\pm}$, and satisfies the following extended Rankine-Hugoniot jump conditions

[TABLE]

for the Bernoulli function $B$ given by

[TABLE]

In (1.3)–(1.4), $[F]_{\Gamma_{s}}$ is defined by $[F({\rm x})]_{\Gamma_{s}}:=F({\rm x})|_{\overline{\Omega^{-}}}-F({\rm x})|_{\overline{\Omega^{+}}}$ for ${\rm x}\in\Gamma_{s}$.

One can easily extend Definition 1.1 to the case of $\Omega\subset\mathbb{R}^{n}$ with $n\geq 3$ through replacing a $C^{1}$ curve $\Gamma_{s}$ and a tangent vector field ${\bm{\tau}}_{s}$ on $\Gamma_{s}$ by a $C^{1}$ (hyper)surface $\Gamma_{s}$ and tangent vector fields $\{{\bm{\tau}}_{s}^{(j)}\}_{j=1}^{n-1}$ on $\Gamma_{s}$ with $\{{\bm{\tau}}_{s}^{(j)}\}_{j=1}^{n-1}$ being linearly independent at each point on $\Gamma_{s}$, respectively.

Definition 1.2** (Admissibility of a shock solution).**

Let $(\rho_{\pm},{{\bf u}}_{\pm},p_{\pm},\Phi_{\pm})$ denote $(\rho,{{\bf u}},p,\Phi)$ restricted on $\Omega^{\pm}$, respectively. A shock solution ${\bf U}$ with a shock $\Gamma_{s}$ is said physically admissible if

[TABLE]

for $\ln S=\frac{p}{\rho^{\gamma}}$.

Depending on the Mach number $M=\frac{|{{\bf u}}|}{c(\rho,S)}$ for $c(\rho,S)=\sqrt{\gamma e^{S}\rho^{\gamma-1}}$, analytic and physical features of (1.1) vary. If $M>1$, then the corresponding state ${\bf U}=(\rho,{{\bf u}},p,\Phi)$ is called supersonic. If $M<1$, on the other hand, then the corresponding state is called subsonic. Here, $c(\rho,S)$ is called the local sound speed.

Definition 1.3** (Transonic shock solution).**

A shock $\Gamma_{s}$ is called a transonic shock if $M$ jumps from the state of $M>1$ to the state of $M<1$ across $\Gamma_{s}$.

The goal of this work is to show that, given constant data of $(\rho_{0},-u_{0}{\bf e}_{r},p_{0})$ and electric force $\nabla\Phi=-E_{0}{\bf e}_{r}$ for supersonic flow at the entrance $\Gamma_{\rm{ent}}$, and constant pressure $p_{\rm ex}$ for subsonic flow at the exit $\Gamma_{\rm{ex}}$, the system (1.1) admits a unique radial transonic shock solution in a two dimensional annulus of finite radius, provided that $\rho_{0}$, $u_{0}$, $p_{0}$, $E_{0}$ and $p_{\rm{ex}}$ are positive, and that $E_{0}$ is sufficiently large depending on $(\rho_{0},u_{0},p_{0})$ and the length of the nozzle. See Figure 1.

In [8], by nonlinear ODE analysis of Euler system in two different domains (a flat nozzle, and a divergent nozzle) in $\mathbb{R}^{2}$, it is shown that the geometry of a domain plays a key role to decide analytic behavior of transonic shock solutions. In a divergent nozzle, for fixed entrance data of supersonic flow and for fixed constant pressure at the exit, it is proved that steady Euler system

[TABLE]

admits a unique radial transonic shock solution, provided that the exit pressure is in a certain range for subsonic flow. In a flat nozzle, on the other hand, a transonic shock problem of Euler system with constant boundary data has either no solution or infinitely many solutions. See Figure 2. Overall, the geometry of a domain determines whether a transonic shock problem of Euler system with fixed exit pressure is well-posed or not.

Interestingly, [6] and [7] reveal that if the background charge density $b$ is less than the sonic density, then one dimensional transonic shock solutions of steady isentropic Euler-Poisson system defined in a flat nozzle have a similar feature to radial transonic shock solutions of Euler system in a divergent nozzle. More specifically, for fixed constant entrance data for supersonic flow and constant pressure for subsonic flow at the exit in a flat nozzle, steady isentropic Euler-Poisson system admits a unique one dimensional transonic shock solution, provided that the electric field at the entrance acts in the same direction as the entrance flow velocity. Also, one can directly check that the same is true for the system (1.1). See Figure 3. This conditional well-posedness of a transonic shock problem of Euler-Poisson system with a fixed exit subsonic pressure shows that the self-generated electric field $\nabla\Phi$ with $\nabla\Phi\cdot{\bf u}>0$ in the system (1.1) has the same effect of the geometry of a divergent nozzle in Figure 2. This naturally raises a question on the well-posedness of a transonic shock problem of compressible Euler-Poisson system in a convergent nozzle with a fixed exit subsonic pressure under a strong effect of electric field. And, it is our goal to establish the well-posedness of radial transonic shock problem of (1.1) in a convergent domain under a strong effect of self-generated electric field. If this well-posedness of a transonic shock problem in a convergent nozzle is achieved, then one can expect to establish the dynamical stability of a transonic shock of the system (1.1) in a convergent nozzle. In [5], it has been shown that a one-dimensional transonic shock of Euler system in a convergent nozzle is dynamically unstable. Therefore, the results in this paper show that self-generated force in compressible flow can overcome the geometry of domain to stabilize a certain physical feature of the flow.

2. Radial transonic shock solutions of (1.1) in a convergent domain

Let $(r,\theta)$ be polar coordinates in $\mathbb{R}^{2}$. For fixed constants $r_{0}>r_{1}>0$, define an annular domain

[TABLE]

Set

[TABLE]

In (1.1), assume that $\tilde{b}$ is in $C^{1}(\overline{\mathcal{A}})$ and $\tilde{b}=\tilde{b}(r)$ with

[TABLE]

for some constant $b_{0}>0$. Fix $\gamma>1$. For positive constants $(\rho_{0},u_{0},p_{0},E_{0})$, we prescribe boundary conditions as follows:

[TABLE]

for ${\bf e}_{r}=\frac{\rm x}{|{\rm x}|}$. Since (1.1) and (2.2) are invariant under a coordinate rotation, we set as

[TABLE]

so that (1.1) and (2.2) are rewritten as

[TABLE]

As we seek solutions flowing in the direction of $-{\bf e}_{r}$ in $\mathcal{A}$, it is convenient to introduce new variables

[TABLE]

and to set

[TABLE]

Then (2.3) is equivalent to the following initial value problem for $(\rho,u,p,E)(t)$:

[TABLE]

with

[TABLE]

where we denote $\frac{d}{dt}$ by ′.

The R-H conditions (1.3)–(1.4) for radial solutions of (1.1) in terms of $(\rho,u,p,E)$ become

[TABLE]

where $\Gamma_{s}$ is given as

[TABLE]

From the second and third equation of (2.5), one can directly derive that $S$ from (1.2) satisfies

[TABLE]

This and the first equation in (2.5) yield

[TABLE]

From (2.9) and the definition $M^{2}=\frac{u^{2}}{\gamma p/\rho}$, it is directly derived that

[TABLE]

By (2.9), the third equation in (2.5) can be simplified as

[TABLE]

By $\eqref{m0s0}$ and (2.10), equation $\eqref{bp}$ can be written as a nonlinear equation for $(M,E)$ as follows:

[TABLE]

We solve (2.10) for $\rho$ to get

[TABLE]

then substitute this expression into the last equation in (2.5) to get

[TABLE]

Note that the constants $\mu_{0}$ and $\mu_{1}$ vary depending on $(\rho_{0},u_{0},p_{0})$, but they are independent of $E_{0}$.

By (2.9), (2.12) and (2.14), if $M\neq 1$ and $\rho u>0$ for $0<t<T$, then (2.5) is equivalent to

[TABLE]

which consists of two algebraic equations and a first order nonlinear ODE system for $(M,E)$. In (2.12) and (2.14), we represent $(h_{1},h_{2})$ as functions varying with respect to $\kappa_{0}$. This is because, we will consider the system (2.15) for different values of $\kappa_{0}$, while $m_{0}$ is fixed same always due to the R-H conditions (2.7).

For later use, we also note that (2.15) is equivalent to

[TABLE]

for

[TABLE]

Fix $\gamma>1$, and suppose that the initial condition $({\rho},{u},{p},{E})(0)=(\rho_{0},u_{0},p_{0},E_{0})$ in (2.5) satisfies $M_{0}^{2}:=\frac{u_{0}^{2}}{(\gamma p_{0}/\rho_{0})}>1$. And, suppose that (2.5) has a $C^{1}$ solution $(\rho_{-},u_{-},p_{-},E_{-})$ on the interval $[0,T]$ with

[TABLE]

The unique existence of such a $C^{1}$ solution $(\rho_{-},u_{-},p_{-},E_{-})$ to (2.5) is discussed in the next section.

For a fixed constant $t_{s}(=:r_{0}-r_{s},r_{s}\in(r_{1},r_{0}))\in(0,T)$, we construct a radial shock solution $(\rho,u,p,E)(t;t_{s})$ of (1.1) with incoming state $(\rho_{-},u_{-},p_{-},E_{-})$, and with a shock $\Gamma_{s}=\{t=t_{s}\}$ in the sense of Definition 1.1. For each $t_{s}\in(0,T)$, the shock solution $(\rho,u,p,E)(t;t_{s})$ is represented as

[TABLE]

for $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ defined as follows:

• (1)

Set $(\rho_{s},u_{s},p_{s},E_{s}):=(\rho_{+},u_{+},p_{+},E_{+})(t_{s};t_{s})$. A direct computation using (2.7) yields

[TABLE]

for $K_{-}=\frac{2(\gamma-1)}{\gamma+1}(\frac{1}{2}u_{-}^{2}+\frac{\gamma p_{-}}{(\gamma-1)\rho_{-}})$ where $(\rho_{-},u_{-},p_{-},E_{-})$ are evaluated at $\Gamma_{s}=\{t=t_{s}\}$.

• (2)

Set

[TABLE]

and solve

[TABLE]

with initial condition (2.21) for $t_{s}<t\leq T$. If (2.23) has a unique $C^{1}$ solution with $M\neq 1$ for $t_{s}\leq t\leq T$, we denote the solution by $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ .

By the assumption (2.18), we get from (2.21) that $(\rho_{s},u_{s},p_{s})$ are strictly positive, and that

[TABLE]

for $(M^{2}_{s},M^{2}_{-})(t_{s}):=(\frac{u_{s}^{2}}{(\gamma p_{s}/\rho_{s})},\frac{u_{-}^{2}}{(\gamma p_{-}/\rho_{-})})(t_{s})$. This indicates that, for each $t_{s}\in(0,T)$, the shock solution $(\rho,u,p,E)(t;t_{s})$ is admissible in the sense of Definition 1.2, and that $\Gamma_{s}=\{t=t_{s}\}$ is a transonic shock in the sense of Definition 1.3.

Our main interest is in the radial shock solutions of (1.1) which behave similar to the ones of Euler system in a divergent nozzle(Figure 2). Therefore, we define a particular class of shock solutions as follows:

Definition 2.1** (Radial transonic shock solution of (1.1) with positive direction of electric field).**

For a fixed $t_{s}\in(0,T)$, we define a shock solution $(\rho,u,p,E)(t;t_{s})$ to be a radial transonic shock solution of (1.1) with positive direction of electric field if it satisfies the following conditions:

• (i)

$(\rho,u,p,E)(t;t_{s})$* are strictly positive for $0\leq t\leq T$;*

• (ii)

the Mach number $M_{+}(t;t_{s})=\frac{u_{+}(t;t_{s})}{\sqrt{(\gamma p_{+}(t;t_{s})/\rho_{+}(t;t_{s}))}}$ downstream satisfies

[TABLE]

Our main theorem is stated below.

Theorem 2.2**.**

Fix two constants $\gamma>1$ and $r_{0}>0$. And, fix a $C^{1}$ function $b(t)(=\tilde{b}(r))$ with $b(t)>0$ for $t\in[0,r_{0}]$ and $\|b\|_{C^{1}([0,r_{0}])}\leq b_{0}$ for some constant $b_{0}>0$. Fix positive constant data $(\rho_{0},u_{0},p_{0})$ with $M_{0}(=\sqrt{\frac{u_{0}^{2}}{\gamma p_{0}/\rho_{0}}})>1$.

• (a)

There exists $\underline{E}>0$ depending on $(\gamma,r_{0},b_{0},\rho_{0},u_{0},p_{0})$ such that whenever $E_{0}\geq\underline{E}$, a family of transonic shock solution $(\rho,u,p,E)(t;t_{s})$ in the form of (2.19) is uniquely given on the interval $I_{T}=[0,T]$ with satisfying

[TABLE]

and

[TABLE]

where $T$ is sufficiently small depending on $(\gamma,r_{0},b_{0},\rho_{0},u_{0},p_{0},E_{0})$.

• (b)

For $\gamma\geq 2$, if $r_{1}\in(0,r_{0})$ satisfies $\ln\frac{r_{0}}{r_{1}}<\frac{\gamma+1}{2(\gamma-1)}$, then there exists $E_{*}>0$ depending on $(\gamma,r_{0},r_{1},b_{0},\rho_{0},u_{0},p_{0})$ such that whenever $E_{0}\geq E_{*}$, a family of transonic shock solution $(\rho,u,p,E)(t;t_{s})$ is uniquely given on the interval $I_{T_{*}}:=[0,r_{0}-r_{1}]$ with satisfying (2.26) and (2.27).

Remark 2.3**.**

According to Theorem 2.2, given incoming supersonic radial flow of Euler-Poisson system in an annular domain $\mathcal{A}$, a transonic shock problem with a fixed subsonic pressure on the inner boundary $\Gamma_{\rm ex}$ of $\mathcal{A}$(see Figure 1) is conditionally well-posed in the sense that if the magnitude of the electric field is sufficiently strong, then there exists a unique radial transonic shock solution of (1.1).

The following proposition is the key ingredient to prove Theorem 2.2:

Proposition 2.4**.**

Suppose that a shock solution $(\rho,u,p,E)(t;t_{s})$ in (2.19) is a radial transonic shock solution with positive direction of electric field for all $t_{s}\in[0,T]$, in the sense of Definition 2.1. In addition, assume that

• (i)

$M_{-}^{2}=\frac{u_{-}^{2}}{(\gamma p_{-}/\rho_{-})}$* satisfies*

[TABLE]

• (ii)

for each $t_{s}\in(0,T)$, $\rho_{+}(t;t_{s})$ satisfies

[TABLE]

Then, we have

[TABLE]

Before proving Proposition 2.4, a few preliminary lemmas need to come first.

Lemma 2.5**.**

Under the same assumptions as Proposition 2.4, $\kappa_{s}=\kappa_{s}(t_{s})$ defined by (2.22) satisfies

[TABLE]

Proof.

By (2.21) and (2.22), we get $\kappa_{s}(t_{s})=\left.\frac{\rho_{-}u_{-}^{2}+p_{-}-\rho_{-}K_{-}}{\left({\rho_{-}u_{-}^{2}}/{K_{-}}\right)^{\gamma}}\right|_{t=t_{s}}.$ By using the definition $M_{-}^{2}=\frac{u_{-}^{2}}{(\gamma p_{-}/\rho_{-})}$, we rewrite $\kappa_{s}$ as

[TABLE]

for

[TABLE]

Note that $f(1)=1$, and

[TABLE]

Since $f(M_{-}^{2}(t_{s}))>1$ for all $t_{s}\in(0,T)$ by (2.18), we obtain from (2.31) that

[TABLE]

To prove $\eqref{sprove}$, we differentiate $\eqref{sMain}$ with respect to $t_{s}$ to get

[TABLE]

Then, (2.30) is directly obtained from (2.28) and (2.32).

∎

Under the assumption of (2.18), $(\rho_{s},u_{s},p_{s},E_{s})$ are $C^{1}$ with respect to $t_{s}\in(0,T)$. And, $(h_{1},h_{2})(t,M^{2},E,\kappa_{s})$ (or, $(g_{1},g_{2})(t,\rho,E,\kappa_{s})$) are $C^{1}$ with respect to $(t,M^{2},E,\kappa_{s})$ (or, $(t,\rho,E,\kappa_{s})$), for $(h_{1},h_{2},g_{1},g_{2})$ defined by (2.12), (2.14), (2.17). Therefore, $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ are $C^{1}$ with respect to $t_{s}\in(0,t]$ for each $t\in(0,T)$.

Lemma 2.6**.**

Under the same assumptions as Proposition 2.4, for each fixed $\bar{t}\in(0,T)$, $\rho_{+}(\bar{t};t_{s})$ satisfies

[TABLE]

Proof.

In this proof, we use the second formulation in (2.23).

(Step 1) Fix $\bar{t}\in(0,T]$. For each $t_{s}\in(0,\bar{t}]$, we set as

[TABLE]

By integrating $\frac{d}{d\eta}\rho_{+}(\eta;t_{s})=\mathfrak{g}_{1}(\eta;t_{s})$ with respect to $\eta$ over the interval $[t_{s},t]$ for some $t\in[t_{s},\bar{t}]$, we obtain that

[TABLE]

where $\rho_{s}$ is given in $\eqref{rhc}$. Take the partial derivative of $\rho_{+}(t;t_{s})$ with respect to $t_{s}$ to get

[TABLE]

A direct computation using (2.17) shows that

[TABLE]

By the conditions (2.25) and (2.29), one can choose a positive constant $\lambda_{0}>1$ depending on $(\gamma,\rho_{0},u_{0},p_{0},E_{0},T)$ to satisfy

[TABLE]

for all $t_{s}\in[0,T]$ and $t\in[t_{s},T]$. One can further adjust $\lambda_{0}>0$ so that Lemma 2.5 yields

[TABLE]

(Step 2) Set $X(t;t_{s}):=\frac{\partial\rho_{+}}{\partial t_{s}}(t;t_{s})$. By $\eqref{rho+rs}$, $X$ becomes a solution to

[TABLE]

A direct computation using (2.17), (2.18) and (2.21) yields that

[TABLE]

Since $X(t;t_{s})$ is $C^{1}$ for $t\in[t_{s},T]$, there exists a small constant $\varepsilon>0$ such that $X(t;t_{s})<0$ for all $t\in[t_{s},t_{s}+\varepsilon]$. Let $t^{*}\in(t_{s},\bar{t}]$ be the smallest value of $t$ such that $X(t;t_{s})<0$ for $t<t^{*}$, and $X(t;t_{s})\geq 0$ for $t>t^{*}$. Then,

[TABLE]

(Step 3) For further estimate of $X(t;t_{s})$, we get back to the equation $(\hat{t}E)^{\prime}=g_{2}(t,\rho,E)$ in (2.23). We integrate this equation with respect to $t$ over the interval, then take the partial derivative of the resultant equation with respect to $t_{s}$ to get,

[TABLE]

From this, we obtain that

[TABLE]

By step 2, we get from (2.21), (2.24) and (2.42) that

[TABLE]

By $\eqref{abc}$–$\eqref{xr}$ and $\eqref{e1}$, there exists a constant $\lambda_{1}>0$ such that

[TABLE]

By the simple method of integrating factor, we obtain that

[TABLE]

for $\mu(\eta):=\exp(-\int_{t_{s}}^{\eta}a_{1}(\tilde{\eta};t_{s})\,d\tilde{\eta})$. But this contradicts to (2.40). Therefore, we finally conclude that $X(\bar{t};t_{s})=\frac{\partial\rho_{+}}{\partial t_{s}}(\bar{t};t_{s})<0$. This finishes the proof. ∎

We are now ready to prove the monotonicity of the exit pressure $p_{+}(T;t_{s})$ with respect to $t_{s}$.

Proof of Proposition 2.4.

For $t_{s}\in[0,T]$, define

[TABLE]

Substituting the expressions $u_{+}(t;t_{s})=\frac{m_{0}}{\hat{t}\rho_{+}(t;t_{s})}$ and $\rho_{+}(t;t_{s})=\left(\frac{p_{+}(t;t_{s})}{\kappa_{s}(t_{s})}\right)^{\frac{1}{\gamma}}$ into the definition stated above, we get

[TABLE]

We differentiate (2.44) with respect to $t_{s}$ to get

[TABLE]

with

[TABLE]

From (2.7) and (2.11), it follows that

[TABLE]

By differentiating (2.46) with respect to $t_{s}$, and by applying (2.21), one gets

[TABLE]

Due to Lemma 2.6 and (2.42), it holds that $\frac{\partial E_{+}}{\partial t_{s}}\left(t;t_{s}\right)<0$ for all $t\in\left[t_{s},T\right]$, and this implies that

[TABLE]

Finally, combining this with (2.25) and (2.45) implies (2.27). ∎

3. Supersonic flow

Before proving Theoren 2.2, we first prove the existence of radial supersonic solutions of (1.1) satisfying (2.28).

Proposition 3.1**.**

Fix $\gamma>1$ and $r_{0}>0$ for the definition of the annulus $\mathcal{A}$ in (2.1). Given positive constant data $(m_{0},\kappa_{0},M_{0})$ with $M_{0}>1$, if $r_{1}\in(0,r_{0})$ in (2.1) satisfies

[TABLE]

then there exists a constant $\underline{E}>0$ depending on $(\gamma,m_{0},\kappa_{0},M_{0},r_{0},r_{1},b_{0})$ so that whenever $E_{0}>\underline{E}$, the initial value problem (2.15) with $(M,E)(0)=(M_{0},E_{0})$ has a unique $C^{1}$ solution $(M,E)(t)$ for $t\in[0,r_{0}-r_{1}]$ satisfying (2.28).

Proof.

(Step 1) We rewrite $\eqref{odeM}$ as

[TABLE]

for $K=\frac{2(\gamma-1)}{\gamma+1}B$, where $B$ is defined by (2.6). Since $(h_{1},h_{2})$ from (2.12) and (2.14) are $C^{1}$ with respect to $(t,M^{2},E)$ for $M\neq 1$, the initial value problem (2.15) with $(M,E)(0)=(M_{0},E_{0})$ with $M_{0}>1$ and $E_{0}>0$ has a unique $C^{1}$ solution for $t\in[0,\varepsilon_{0})$ for some small constant $\varepsilon_{0}>0$ with $M^{2}(t)>1$ for all $t\in[0,\varepsilon_{0})$.

(Step 2) By the definition of $K$ and (2.11), $K$ satisfies

[TABLE]

Combine this equation with (2.14) to get

[TABLE]

From (2.14) and $M^{2}>1$, it follows that

[TABLE]

By integrating this inequality over the interval $[0,t]$ for $t\in(0,\varepsilon_{0})$, we obtain that

[TABLE]

for a constant $\alpha_{1}>0$ depending only on $(r_{0},\kappa_{0},m_{0},b_{0},\gamma)$. Note that $\alpha_{1}$ is independent of $\varepsilon_{0}$ and $E_{0}$. Combine (3.6) with (3.4) to get

[TABLE]

for $K_{0}:=\frac{\gamma-1}{\gamma+1}u_{0}^{2}+\frac{2\gamma p_{0}}{(\gamma+1)\rho_{0}}$ and a constant $\alpha_{2}>0$ depending on $(r_{0},r_{1},M_{0},\kappa_{0},m_{0},\gamma,b_{0})$ but independent of $(\varepsilon_{0},E_{0})$.

(Step 3) Fix a constant $r_{1}>0$ to satisfy (3.1), and set

[TABLE]

so that $0<\delta_{0}<1$ holds. Then we have $r_{0}E_{0}(1-\frac{2(\gamma-1)}{\gamma+1}\ln\frac{r_{0}}{r_{0}-\varepsilon_{0}})-\alpha_{2}\geq r_{0}E_{0}\delta_{0}-\alpha_{2}$, whenever $\varepsilon_{0}\in(0,r_{0}-r_{1}).$ Set $\underline{E}$ as

[TABLE]

Note that $\underline{E}$ depends only on $(\gamma,m_{0},\kappa_{0},M_{0},r_{0},r_{1},b_{0})$. If $E_{0}>\underline{E}$, then we obtain that

[TABLE]

(Step 4) Given positive constant data $(m_{0},\kappa_{0},M_{0},E_{0})$ with $M_{0}>1$ and $E_{0}>\underline{E}$, we have shown in (step 1)–(step 3) that the initial value problem (2.15) with $(M,E)(0)=(M_{0},E_{0})$ has a unique smooth solution on the interval $[0,\varepsilon_{0})$ for some small constant $\varepsilon_{0}>0$, furthermore the solution satisfies $(M^{2})^{\prime}>0$ on $(0,\varepsilon_{0})$. It remains to extend the solution up to $t=r_{0}-r_{1}(=:T)$.

By (3.5) and (3.6), $E$ can be extended up to $t=\varepsilon_{0}$ as

[TABLE]

there exists a constant $\alpha_{3}>0$ depending on $(\gamma,m_{0},\kappa_{0},M_{0},E_{0},b_{0},r_{0},r_{1})$, but independent of $\varepsilon_{0}$ such that

[TABLE]

Then (2.12) yields that

[TABLE]

By integrating the inequality above over the interval $[0,t)$ for $t\in(0,\varepsilon_{0})$, we obtain that

[TABLE]

By (3.7) and the monotone convergence theorem, the limit $\underset{t\to\varepsilon_{0}-}{\lim}M^{2}(t)$ exists, therefore $M^{2}$ can be extended up to $t=\varepsilon_{0}$. Then, one can repeat the argument in (step 1)–(step 3) to conclude that the $C^{1}$ solution to (2.15) and $(M,E)(0)=(M_{0},E_{0})$ with $M_{0}>1$ and $E_{0}>\underline{E}$ uniquely exists up to $t=r_{0}-r_{1}$, and the solution satisfies (2.28) for all $t\in(0,r_{0}-r_{1}]$.

∎

Remark 3.2**.**

In the definition (2.1) of $\mathcal{A}$, denote $r_{1}$ by $r_{1}=\lambda r_{0}$ for $\lambda\in(0,1)$. According to Proposition 3.1, if the convergent ratio of the annulus $\lambda(=\frac{r_{1}}{r_{0}})$ is greater than $e^{-\frac{\gamma+1}{2(\gamma-1)}}$, then a radial supersonic solution to (1.1) in $\mathcal{A}$ is relatively accelerating throughout the annulus $\mathcal{A}$, in the sense that the Mach number $M$ monotonically increases in the flow direction, provided that the magnitude of electric field ${\bf E}=-E_{0}\hat{{\bf r}}$ at the entrance $\Gamma_{0}$ is sufficiently large depending on $(\gamma,m_{0},\kappa_{0},M_{0},r_{0},b_{0},\lambda)$. Since $\underset{\gamma\to 1+}{\lim}e^{-\frac{\gamma+1}{2(\gamma-1)}}=0$, this conditional monotonicity property holds for a bigger class of annuli as $\gamma$ is close to $1$.

4. Proof of Theorem 2.2

This section is devoted to prove our main theorem.

Proof of Theorem 2.2 (a).

(Step 1) Fix $\gamma>1$, $r_{0}>0$, then fix $\tilde{r}_{1}\in(0,r_{0})$ to satisfy $\ln\frac{r_{0}}{\tilde{r}_{1}}<\frac{\gamma+1}{2(\gamma-1)}$. Given positive constant $(m_{0},\kappa_{0},M_{0})$ with $M_{0}>1$, let $\underline{E}$ be from Proposition 3.1 for $r_{1}$ replaced by $\tilde{r}_{1}$. For $E_{0}>\underline{E}$, let $(\rho_{-},u_{-},p_{-},E_{-})(t)$ be the solution to the initial value problem (2.15) with $(M_{-},E_{-})(0)=(M_{0},E_{0})$. By Proposition 3.1, (2.15) with $(M_{-},E_{-})(0)=(M_{0},E_{0})$ has a unique solution in $[0,r_{0}-\tilde{r}_{1}]$, and the solution $(\rho_{-},u_{-},p_{-},E_{-})(t)$ satisfies

[TABLE]

Fix $t_{s}\in[0,T_{1}]$ for $T_{1}=r_{0}-\tilde{r}_{1}$, and let $(\rho_{s},u_{s},p_{s},E_{s})$ be given by (2.21). Set $\kappa_{s}$ as in (2.22), and consider the following initial value problem

[TABLE]

for $(g_{1},g_{2})$ defined by (2.17). Set $M_{s}:=u_{s}/\sqrt{\frac{\gamma p_{s}}{\rho_{s}}}=u_{s}/\sqrt{\gamma\kappa_{s}\rho_{s}^{\gamma-1}}$, then (2.24) implies that $g_{1}$ and $g_{2}$ are $C^{1}$ with respect to $(t,\rho,E)$ near $(t_{s},\rho_{s},E_{s})$. Therefore, (4.1) has a unique $C^{1}$ solution $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ with $\rho_{+}>0$ for $t\in(t_{s}-\varepsilon_{0},t_{s}+\varepsilon_{0})$ for some small constant $\varepsilon_{0}>0$. Furthermore, the solution is subsonic in the sense that $\gamma\kappa_{s}\rho_{+}^{\gamma-1}-\frac{m_{0}^{2}}{\hat{t}^{2}\rho_{+}^{2}}>0$, or equivalently $M_{+}^{2}-1<0$ on $(t_{s}-\varepsilon_{0},t_{s}+\varepsilon_{0})$ for $M_{+}^{2}=\frac{u_{+}^{2}}{\gamma\kappa_{s}\rho_{+}^{\gamma-1}}$.

Moreover, the solution $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ satisfies

[TABLE]

This can be checked as follows: By (2.21), (2.23), (2.24), (3.2) and Proposition 3.1, we have

[TABLE]

Then, (4.2) follows from (2.13) combined with the inequality right above. By (4.2), there exists a small constant $\varepsilon_{1}\in(0,\varepsilon_{0})$ depending on $(\gamma,m_{0},\kappa_{0},M_{0},E_{0},r_{0},t_{s},b_{0})$ such that $(M_{+}^{2})^{\prime}(t;t_{s})<0$ thus $\rho_{+}^{\prime}(t;t_{s})>0$ hold by (2.13) on $[t_{s}-\varepsilon_{1},t_{s}+\varepsilon_{1}]$.

(Step 2) For each $t_{s}\in[0,r_{0}-\tilde{r}_{1}]$, let $l(t_{s})>0$ be given to satisfy the following properties:

• (i)

The initial value problem (4.1) is uniquely solvable for $t_{s}\leq t<l(t_{s})$;

• (ii)

[TABLE]

for $t_{s}\leq t<l(t_{s})$;

• (iii)

(ii) does not hold for $t\geq l(t_{s})$.

Note that we may have $l(t_{s})=\infty$. Choose $T$ as

[TABLE]

Such a constant $T$ is strictly positive. Then, we obtain from Proposition 2.4 that $\frac{dp_{+}}{dt_{s}}(T;t_{s})<0$ for all $t_{s}\in(0,r_{0}-r_{1})$. This proves Theorem 2.2 (a). ∎

Remark 4.1**.**

If $1<\gamma<2$, a lengthy computation shows the initial value problem $\eqref{ivp-ds-sub}$ is uniquely solvable only on a finite interval especially when $E_{s}>0$ is sufficiently large, which can happen if $E_{0}>0$ is sufficiently large. Furthermore, life-span of $C^{1}$ solution $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ to (4.1) for $t\geq t_{s}$ converges to [math] as $E_{s}\to\infty$. For $\gamma\geq 2$, however, one can construct a family of radial transonic shock solutions satisfying (2.27) in an annular domain $\mathcal{A}$ given by (2.1) whenever (3.1) holds.

Proof of Theorem 2.2 (b).

(Step 1) Fix $\gamma\geq 2$, and fix two constants $r_{0}>r_{1}>0$ with $\ln\frac{r_{0}}{r_{1}}<\frac{2(\gamma-1)}{\gamma+1}$ to define an annulus $\mathcal{A}$ by (2.1). Given positive constant $(m_{0},\kappa_{0},M_{0})$ with $M_{0}>1$, let $\underline{E}$ be from Proposition 3.1. For $E_{0}>\underline{E}$ to be further specified later, let $(\rho_{-},u_{-},p_{-},E_{-})(t)$ be the solution to the initial value problem (2.15) with $(M_{-},E_{-})(0)=(M_{0},E_{0})$. By Proposition 3.1, the solution $(\rho_{-},u_{-},p_{-},E_{-})(t)$ satisfies

[TABLE]

Differently from the proof of Theorem 2.2(a), a lower bound of $E_{0}$ needs to be adjusted further to acquire a family of radial transonic shock solutions satisfying (2.27) for $t_{s}\in(0,T_{*})$ for $T_{*}=r_{0}-r_{1}$.

(Step 2) Fix $t_{s}\in(0,T_{*})$, and let $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ be a $C^{1}$ solution to (4.1). As discussed in the proof of Theorem 2.2(a), $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ is well defined on $(t_{s}-\varepsilon_{0},t_{s}+\varepsilon_{0})$ with satisfying (4.3) for some small $\varepsilon_{0}>0$. In this step, we find a sufficient condition for $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ to satisfy

[TABLE]

for any $t_{s}\in(0,T_{*})$.

Rewrite $\rho^{\prime}=g_{1}(t,\rho,E,\kappa_{s})$ in $\eqref{def-gs}$ as

[TABLE]

One can see from $\eqref{oderhorho}$ that $\eqref{sub-condition}$ is equivalent to $\rho_{+}(t;t_{s})>0$, $\hat{t}E_{+}(t;t_{s})-\frac{m_{0}^{2}}{\hat{t}^{2}\rho_{+}^{2}(t;t_{s})}>0$ and $0<M_{+}(t;t_{s})<1$ for $t\in[t_{s},r_{0}-r_{1}]$.

Integrate the differential inequality $(\hat{t}E)^{\prime}\geq-\hat{t}b_{0}$ over the interval $[0,t]$ to get

[TABLE]

for some constant $\beta_{1}>0$ depending only on $(r_{0},r_{1},b_{0})$.

Setting as $\hat{t}_{s}=r_{0}-t_{s}$, the estimate $\eqref{inequality-lb-Eds}$ yields

[TABLE]

If

[TABLE]

then we have

[TABLE]

for some constant $\delta_{s}\in(0,1)$. This can be checked as follows: We rewrite $\eqref{odeM}$ as

[TABLE]

for $\mu_{s}:=(\gamma+1)\frac{1}{\gamma\kappa_{s}}\left(\frac{\gamma\kappa_{s}}{m_{0}^{2}}\right)^{\frac{\gamma-1}{\gamma+1}}$. If $F_{s}(t_{s})\geq\beta_{1}$, then (4) implies that

[TABLE]

thus $M_{+}^{2}$ decreases when $M_{+}^{2}$ is less than $1$, and sufficiently close to 1. This proves (4.9).

Next, we show that

[TABLE]

Since $\rho_{+}(t;t_{s})$ monotonically increases over the interval $(t_{s},t_{s}+\varepsilon_{0})$, by using (2.17), (2.23), (4.5) and (4.9), we get

[TABLE]

for some constant $C_{s}>0$ depending on $(\gamma,r_{0},r_{1},t_{s},\rho_{s})$. Here, we used the fact that $\hat{t}E_{+}(t;t_{s})\leq\hat{t}_{s}E_{s}+\rho_{+}(t;t_{s})\int_{t_{s}}^{t}(r_{0}-\eta)\,d\eta$ provided that $\rho_{+}$ monotonically increases up to $t$. This is obtained from the equation $(\hat{t}E_{+})^{\prime}=\hat{t}(\rho_{+}-b(t))<\hat{t}\rho_{+}$. Since $\gamma\geq 2$, (4.11) is obtained from (4.12) and the monotone sequence theorem. In (4) and (4.9), the estimates are independent of $\varepsilon_{0}$. And, (4.12) shows that $\rho_{+}(t;t_{s})$ does not blow up for $t\leq T_{*}(<r_{0})$ when $\rho_{+}$ monotonically increases. Therefore, by the method of continuation, the following lemma is obtained:

Lemma 4.2**.**

For a fixed $t_{s}\in(0,T_{*})$, if the condition (4.8) holds, then the corresponding subsonic solution $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ is well defined on the interval $[t_{s},T_{*}]$, and it satisfies the properties (4.4).

(Step 3) By Proposition 2.4 and Lemma 4.2, if we find $E_{*}\in[\underline{E},\infty)$ depending only on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0})$ to satisfy $F_{s}(t_{s})\geq\beta_{1}$, then Theorem 2.2(b) is proved. The rest of the proof is devoted to find such a constant $E_{*}$.

Lemma 4.3**.**

For each $t_{s}\in(0,T_{*})$, set $q_{s}:=\frac{r_{0}}{\hat{t}_{s}}$. For any given $\lambda>0$ and $\sigma>0$, there exists a constant $\underline{E}_{*}\in[\underline{E},\infty)$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0})$ and on $(t_{s},\lambda,\sigma)$ such that, whenever $E_{0}\geq\underline{E}_{*}$, the solution to $\eqref{system-simplified1}$ with $(M_{-},E_{-})(0)=(M_{0},E_{0})$ satisfies

[TABLE]

Proof.

By Proposition 3.1, for $E_{0}>\underline{E}$, the solution to $\eqref{system-simplified1}$ uniquely exists and satisfies $M_{-}>1$ and $M_{-}^{\prime}>0$ in $[0,T_{*}]$. Then, by (2.14) and $\eqref{kprime}$, we obtain that

[TABLE]

Combine this with (3.6) to have

[TABLE]

for all $t\in[0,t_{s}]$, where a constant $\alpha_{2}>0$ is chosen depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0})$. Since $1-\frac{\ln q_{s}}{\ln(q_{s}+\sigma)}>0$, one can choose a constant $\underline{E}_{*}\in[\underline{E},\infty)$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0})$ and on $(t_{s},\lambda,\sigma)$ such that (4.14) implies (4.13) whenever $E_{0}\geq\underline{E}_{*}$. ∎

(Step 4) A direct computation with using $\eqref{rhc}$ yields that

[TABLE]

Combine this expression with $u_{-}^{2}<\frac{\gamma+1}{\gamma-1}K_{-}$, $E_{s}=E_{-}(t_{s})$, and $\ln\frac{r_{0}}{r_{1}}<\frac{2(\gamma-1)}{\gamma+1}$ to get

[TABLE]

for $F_{s}(t_{s})$ from (4.5).

By $\eqref{teu2-estimate-crutial}$ and $\eqref{tses1}$, if $G_{s}(t_{s})>0$ for all $t_{s}\in(0,T_{*})$, then $(\rho_{+},u_{+},p_{+},E_{+})(t;t_{s})$ satisfies (4.3) for $t_{s}\leq t<T_{*}$, for each $t_{s}\in(0,T_{*})$ so that Theorem 2.2(b) is proved.

It remains to find $\underline{E}_{\flat}\in[\underline{E},\infty)$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0})$ so that $G_{s}(t_{s})>0$ for all $t_{s}\in(0,t_{*})$, whenever $E_{0}\geq\underline{E}_{\flat}$. To find such a constant $\underline{E}_{\flat}$, the following lemma is needed.

Lemma 4.4**.**

For $\gamma\geq 2$, it holds that

[TABLE]

for any $\xi\in[1,e^{\frac{\gamma+1}{2(\gamma-1)}})$.

Proof.

Define $f(\xi):=2(\frac{\gamma-1}{\gamma+1})^{2}{e^{\frac{\gamma+1}{(\gamma-1)}}}\frac{\ln\xi}{\xi^{2}}$ for $\xi>0$. Since $f^{\prime}(\xi)=2e^{\frac{\gamma-1}{\gamma+1}}(\frac{\gamma-1}{\gamma+1})^{2}\frac{1}{\xi^{3}}(1-2\ln\xi),$ the maximum value of $f$ is acquired at $\xi=e^{\frac{1}{2}}$, that is, $f(\xi)\leq f(e^{\frac{1}{2}})=e^{\frac{2}{\gamma-1}}(\frac{\gamma-1}{\gamma+1})^{2}$ for $\xi>0$. Therefore, Lemma 4.4 is proved if we show that $e^{\frac{2}{\gamma-1}}(\frac{\gamma-1}{\gamma+1})^{2}<1$ for all $\gamma\geq 2$.

Set $g(\gamma):=e^{\frac{1}{\gamma-1}}(\frac{\gamma-1}{\gamma+1})$, then we have

[TABLE]

This implies that $f(e^{\frac{1}{2}})=g(\gamma)^{2}<1$ for $\gamma\geq 2$.

∎

(Step 5) Suppose that $t_{s}\in[\zeta,T_{*})$ for some $\zeta\in(0,T_{*})$.

Claim. There exists a constant $\underline{E}^{*}_{(\zeta)}$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0},\zeta)$ such that whenever $E_{0}\geq\underline{E}^{*}_{(\zeta)}$, we have

[TABLE]

*for the constant $\beta_{1}>0$ from (4.6). *

By Lemma 4.4, there exist constants $\sigma_{\gamma}>0$ and $\underline{M}_{\gamma}>1$ depending only on $\gamma$ such that

[TABLE]

One can choose $E^{(1)}_{\zeta}\in[\underline{E},\infty)$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0},\zeta)$ so that whenever $E_{0}>E^{(1)}_{\zeta}$, the solution to $\eqref{system-simplified1}$ with $(M_{-},E_{-})(0)=(M_{0},E_{0})$ satisfies

[TABLE]

This can be verified as follows: Following the proof of Proposition 3.1, one can directly check that, for any constant $\lambda_{1}>0$, there exists a constant $\underline{E}_{\lambda_{1}}\in[\underline{E},\infty)$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0},\lambda_{1})$ such that, whenever $E_{0}\geq\underline{E}_{\lambda_{1}}$, the solution to $\eqref{system-simplified1}$ with $(M_{-},E_{-})(0)=(M_{0},E_{0})$ satisfies $\hat{t}E_{-}-K_{-}>\lambda_{1}$ for all $t\in[0,T_{*}]$. Since, for $E_{0}>\underline{E}$, $M_{-}^{\prime}>0$ in $[0,r_{0}-r_{1}]$, by $\eqref{equation-M-another}$,

[TABLE]

for a constant $\mu_{2}>0$ depending only on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0})$. By integrating this inequality over the interval $[0,\zeta]$, it is obtained that

[TABLE]

therefore, one can choose $\lambda_{1}>0$ large depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},\zeta)$ to satisfy (4.19). For such $\lambda_{1}$, we choose as $E^{(1)}_{\zeta}=\underline{E}_{\lambda_{1}}$.

By (4.18) and (4.19), if $E_{0}\geq E^{(1)}_{\zeta}$, then we get

[TABLE]

We apply Lemma 4.3 with choosing $\lambda=2\beta_{1}$ to conclude that there exists a constant $E^{(2)}_{\zeta}\in[E^{(1)}_{\zeta},\infty)$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0},\zeta)$ such that

[TABLE]

By choosing as $\underline{E}^{*}_{(\zeta)}=E^{(2)}_{\zeta}$, Claim is verified.

(Step 6) Claim: One can choose constants $\zeta_{*}\in(0,T_{*})$ and $E^{(3)}_{\zeta_{*}}\in[\underline{E},\infty)$ depending on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0})$ so that whenever $E_{0}\geq E^{(3)}_{\zeta_{*}}$, we have

[TABLE]

For $q_{s}=\frac{r_{0}}{r_{0}-t_{s}}\in(1,\frac{r_{0}}{r_{0}-\zeta_{*}}]$, or equivalently $t_{s}\in(0,\zeta_{*}]$, we have

[TABLE]

Choose $\zeta_{*}\in(0,T_{*})$ sufficiently small to satisfy $\frac{1}{2\ln\left(\frac{r_{0}}{r_{0}-\zeta_{*}}\right)}>e^{\frac{\gamma+1}{\gamma-1}}\left(\frac{\gamma-1}{\gamma+1}+\frac{2}{(\gamma+1)M_{0}^{2}}\right)$, then choose a constant $\sigma_{\zeta_{*}}>0$ to satisfy $\frac{1}{2\ln\left(\frac{r_{0}}{r_{0}-\zeta_{*}}+\sigma_{\zeta_{*}}\right)}>e^{\frac{\gamma+1}{\gamma-1}}\left(\frac{\gamma-1}{\gamma+1}+\frac{2}{(\gamma+1)M_{0}^{2}}\right)$. Such a constant $\zeta_{*}$ can be chosen depending on $(\gamma,r_{0},\rho_{0},u_{0},p_{0})$, and so does $\sigma_{\zeta_{*}}$. By (4.22), the choice of $\zeta_{*}$, and Lemma 4.3, there exists a constant $E^{(3)}_{\zeta_{*}}\in[\underline{E},\infty)$ depending only on $(\gamma,r_{0},r_{1},\rho_{0},u_{0},p_{0},b_{0})$ so that whenever $E_{0}\geq E^{(3)}_{\zeta_{*}}$, we have

[TABLE]

The claim is verified.

(Step 7) Finally, we choose $E_{*}$ for Theorem 2.2(b) as

[TABLE]

for $\underline{E}^{*}_{(\zeta)}$ from step 5, so that the function $G_{s}(t_{s})$ from (4.15) satisfies $G_{s}(t_{s})\geq\beta_{1}$ for all $t_{s}\in(0,T_{*})$, thus the condition (4.8) holds for all $t_{s}\in(0,T_{*})$ whenever $E_{0}\geq E_{*}$.

This finishes the proof of Theorem 2.2 (b). ∎