# Structural Stability of Transonic Shock Flows with an External Force

Shangkun Weng  School of Mathematics and Statistics, Wuhan University, Wuhan,
Hubei Province, 430072, People’s Republic of China<EMAIL_ADDRESS>and
Wengang Yang  School of Mathematics and Statistics, Wuhan University, Wuhan,
Hubei Province, 430072, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the structural stability of a transonic shock passing
through a flat nozzle for two-dimensional steady compressible flows with an
external force. We first establish the existence and uniqueness of one
dimensional transonic shock solutions to the steady Euler system with an
external force by prescribing suitable pressure at the exit of the nozzle when
the upstream flow is a uniform supersonic flow. It is shown that the external
force helps to stabilize the transonic shock in flat nozzles and the shock
position is uniquely determined. Then we are concerned with the structural
stability of these transonic shock solutions when the exit pressure is
suitably perturbed. One of the new ingredients in our analysis is to use the
deformation-curl decomposition to the steady Euler system developed in [27] to
deal with the transonic shock problem.

###### Key words and phrases:

transonic shock, stabilization effect of external force, structural stability,
deformation-curl decomposition.

###### 2010 Mathematics Subject Classification:

76H05, 35M12, 35L65, 76N15

##  1\. Introduction and main results

The studies of transonic shock solutions for inviscid compressible flows in
different kinds of nozzles had a long history and had obtained many important
new achievements during the past twenty years. Courant and Friedrichs [9] had
described the transonic shock phenomena in a de Laval nozzle whose cross
section decreases first and then increases. It was observed in experiment that
if the upcoming flow becomes supersonic after passing through the throat of
the nozzle, to match the prescribed appropriately large exit pressure, a shock
front intervenes at some place in the diverging part of the nozzle and the gas
is compressed and slowed down to subsonic speed.

People first used the quasi one dimensional model to study the transonic shock
problem [1, 9, 10, 14]. The structural stability of multidimensional transonic
shocks in flat or diverging nozzles were further investigated in [6, 30, 31]
using the steady potential flows with different kinds of boundary conditions.
In particular, [30, 31] proved that the stability of transonic shocks for
potential flows is usually ill-posed under the perturbations of the exit
pressure. Many researchers also considered the transonic shock problem in the
flat or almost flat nozzles with the exit pressure satisfying some special
constraint, see [3, 4, 5, 15, 29] and the references therein. Recently, there
is an interesting progress on the stability and existence of transonic shock
solutions to the two dimensional and three dimensional axisymmetric steady
compressible Euler system in an almost flat finite nozzle with the receiver
pressure prescribed at the exit of the nozzle (see [11, 12]), where the shock
position was uniquely determined.

The structural stability of the transonic shock problem in two dimensional
divergent nozzles under the perturbations for the exit pressure was first
established in [16] when the opening angle of the nozzle is suitably small.
Later on, this restriction was removed in [17, 20]. Furthermore, the transonic
shock in general two dimensional straight divergent nozzles was shown in [20]
to be structurally stable under generic perturbations for both the nozzle
shape and the exit pressure. The existence and stability of transonic shock
for three dimensional axisymmetric flows without swirl in a conic straight
nozzle were established in [18, 19] with respect to small perturbations of the
exit pressure. For the structural stability under the axisymmetric
perturbation of the nozzle wall, a modified Lagrangian coordinate was
introduced in [26] to deal with the corner singularities near the intersection
points of the shock surface and nozzle boundary and the artificial singularity
near the axis simultaneously. Most recently, the authors in [24, 25] studied
radially symmetric transonic flow with/without shock in an annulus. Thanks to
the effect of angular velocity, it was found in [24] that besides the well-
known supersonic-subsonic shock in a divergent nozzle as in the case without
angular velocity, there exists a supersonic-supersonic shock solution, where
the downstream state may change smoothly from supersonic to subsonic.
Furthermore, there exists a supersonic-sonic shock solution where the shock
circle and the sonic circle coincide.

In this paper, we will consider similar transonic shock phenomena occurring in
a flat nozzle when the fluid is exerted with an external force. The 2-D steady
compressible isentropic Euler system with external force are of the form

(1.1) $\displaystyle\begin{cases}\partial_{x_{1}}(\rho
u_{1})+\partial_{x_{2}}(\rho u_{2})=0,\\\ \partial_{x_{1}}(\rho
u_{1}^{2}+P(\rho))+\partial_{x_{2}}(\rho
u_{1}u_{2})=\rho\partial_{x_{1}}\Phi,\\\ \partial_{x_{1}}(\rho
u_{1}u_{2})+\partial_{x_{2}}(\rho
u_{2}^{2}+P(\rho))=\rho\partial_{x_{2}}\Phi,\end{cases}$

where $(u_{1},u_{2})={\bf u}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is the
unknown velocity filed and $\rho:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is the
density, and $\Phi(x_{1},x_{2})$ is a given potential function of external
force. For the ideal polytropic gas, the equation of state is given by
$P(\rho)=A\rho^{\gamma}$, here $A$ and $\gamma$ ($1<\gamma<3$) are positive
constants. We take $A=1$ throughout this paper for the convenience.

To this end, let’s firstly focus on the 1-D steady compressible flow with an
external force on an interval $I=[L_{0},L_{1}]$, which is governed by

(1.2) $\displaystyle\begin{cases}(\bar{\rho}\bar{u})^{\prime}(x_{1})=0,\\\
\bar{\rho}\bar{u}\bar{u}^{\prime}+\frac{d}{dx_{1}}P(\bar{\rho})=\bar{\rho}\bar{f}(x_{1}),\\\
\bar{\rho}(L_{0})=\rho_{0}>0,\ \ \bar{u}(L_{0})=u_{0}>0,\end{cases}$

where we assume that the flow state at the entrance $x_{1}=L_{0}$ is
supersonic, meaning that
$u_{0}^{2}>c^{2}(\rho_{0})=\gamma\rho_{0}^{\gamma-1}$.

Denote $J=\bar{\rho}\bar{u}=\rho_{0}u_{0}>0$, then it follows from (1.2) that

(1.3)
$\displaystyle\begin{cases}\bar{\rho}(x_{1})=\frac{J}{\bar{u}(x_{1})},\\\
((\bar{u})^{\gamma+1}-\gamma
J^{\gamma-1})\bar{u}^{\prime}=\bar{u}^{\gamma}\bar{f}.\end{cases}$

Also one has

(1.4)
$\displaystyle\bar{u}^{\prime}=\frac{\bar{u}\bar{f}}{\bar{u}^{2}-c^{2}(\bar{\rho})},\quad\bar{\rho}^{\prime}=-\frac{\bar{\rho}\bar{f}}{\bar{u}^{2}-c^{2}(\bar{\rho})},$
(1.5)
$\displaystyle\frac{d}{dx_{1}}\bar{M}^{2}(x_{1})=\frac{(\gamma+1)\bar{M}^{2}}{\bar{M}^{2}-1}\frac{\bar{f}}{c^{2}(\bar{\rho})},$

where $\bar{M}(x_{1})=\frac{\bar{u}(x_{1})}{c(\bar{\rho})}$ is the Mach
number.

Since $\bar{M}^{2}(L_{0})>1$, it follows from (1.5) that if the external force
satisfies

(1.6) $\displaystyle\bar{f}(x_{1})>0,\ \ \forall L_{0}<x_{1}<L_{1},$

then the problem (1.2) has a global supersonic solution
$(\bar{\rho}^{-},\bar{u}^{-})$ on $[L_{0},L_{1}]$. If one prescribes a large
enough end pressure at $x_{1}=L_{1}$, a shock will form at some point
$x_{1}=L_{s}\in(L_{0},L_{1})$ and the gas is compressed and slowed down to
subsonic speed, the gas pressure will increase to match the given end
pressure. Mathematically, one looks for a shock $x_{1}=L_{s}$ and smooth
functions $(\bar{\rho}^{\pm},\bar{u}^{\pm},\bar{P}^{\pm})$ defined on
$I^{+}=[L_{s},L_{1}]$ and $I^{-}=[L_{0},L_{s}]$ respectively, which solves
(1.3) on $I^{\pm}$ with the jump at the shock $x_{1}=L_{s}\in(L_{0},L_{1})$
satisfying the physical entropy condition
$[\bar{P}(L_{s})]=\bar{P}^{+}(L_{s})-\bar{P}^{-}(L_{s})>0$ and the Rankine-
Hugoniot conditions

(1.7) $\displaystyle\begin{cases}[{\bar{\rho}}{\bar{u}}](L_{s})=0,\\\
[{\bar{\rho}}{\bar{u}}^{2}+P({\bar{\rho}})](L_{s})=0.\end{cases}$

and also the boundary conditions

(1.8) $\displaystyle\rho(L_{0})=\rho_{0},\ u(L_{0})=u_{0}>0,$ (1.9)
$\displaystyle\bar{P}(L_{1})=P_{e}.$

We will show that there is a unique transonic shock solution to the 1-D Euler
system when the end pressure $P_{e}$ lies in a suitable interval. Such a
problem will be solved by a shooting method employing the monotonicity
relation between the shock position and the end pressure.

######  Lemma 1.1.

Suppose that the initial state $(u_{0},\rho_{0})$ at $x_{1}=L_{0}$ is
supersonic and the external force $f$ satisfying (1.6), there exists two
positive constants $P_{0},P_{1}>0$ such that if the end pressure
$P_{e}\in(P_{1},P_{0})$, there exists a unique transonic shock solution
$(\bar{u}^{-},\bar{\rho}^{-})$ and $(\bar{u}^{+},\bar{\rho}^{+})$ defined on
$I^{-}=[L_{0},L_{s})$ and $I^{+}=(L_{s},L_{1})$ respectively, with a shock
located at $x_{1}=L_{s}\in(L_{0},L_{1})$. In addition, the shock position
$x_{1}=L_{s}$ increases as the exit pressure $P_{e}$ decreases. Furthermore,
the shock position $L_{s}$ approaches to $L_{1}$ if $P_{e}$ goes to $P_{1}$
and $L_{s}$ tends to $L_{0}$ if $P_{e}$ goes to $P_{0}$.

###### Proof.

The existence and uniqueness of smooth supersonic flow
$(\bar{u}^{-},\bar{\rho}^{-})$ starting from $(\rho_{0},u_{0})$ on
$[L_{0},L_{1}]$ is trivial. Suppose the shock occurs at
$x_{1}=L_{s}\in(L_{0},L_{1})$, then it is well-known that there exists a
unique subsonic state $(\bar{u}^{+}(L_{s}),\bar{\rho}^{+}(L_{s}))$ satisfying
the Rankine-Hugoniot conditions (1.7) and the entropy condition. With
$(\bar{u}^{+}(L_{s}),\bar{\rho}^{+}(L_{s}))$ as the initial data, the equation
(1.2) has a unique smooth solution $(\bar{u}^{+},\bar{\rho}^{+})$ on
$[L_{s},L_{1}]$. Denote $p_{e}=(\bar{\rho}^{+}(L_{1}))^{\gamma}$. In the
following, we show that the monotonicity between the shock position
$x_{1}=L_{s}$ and the exit pressure $P_{e}=(\bar{\rho}^{+}(L_{1}))^{\gamma}$.
$\bar{\rho}^{+}(L_{1})$ is regarded as a function of $L_{s}$. Since
$(\bar{\rho}^{+}\bar{u}^{+})(L_{s})=(\bar{\rho}^{-}\bar{u}^{-})(L_{s})=J=\rho_{0}u_{0}>0$,
then

(1.10)
$\displaystyle\bar{u}^{-}(L_{s})+\frac{J^{\gamma-1}}{(\bar{u}^{-}(L_{s}))^{\gamma}}=\bar{u}^{+}(L_{s})+\frac{J^{\gamma-1}}{(\bar{u}^{+}(L_{s}))^{\gamma}}.$

It follows from the second equation in (1.2) that

$\displaystyle\frac{1}{2}(\bar{u}^{+}(L_{1}))^{2}+\frac{\gamma}{\gamma-1}({\bar{\rho}}^{+}(L_{1}))^{\gamma-1}-\bar{\Phi}(L_{1})=\frac{1}{2}(\bar{u}^{+}(L_{s}))^{2}+\frac{\gamma}{\gamma-1}(\bar{\rho}^{+}(L_{s}))^{\gamma-1}-\bar{\Phi}(L_{s}).$

Differentiating with respect to $L_{s}$, one deduces that

(1.11)
$\displaystyle\left(\gamma(\bar{\rho}^{+}(L_{1}))^{\gamma-2}-\frac{J^{2}}{(\bar{\rho}^{+}(L_{1}))^{3}}\right)\frac{d\bar{\rho}^{+}(L_{1})}{dL_{s}}$
$\displaystyle=\left(\gamma(\bar{\rho}^{+}(L_{s}))^{\gamma-2}-\frac{J^{2}}{(\bar{\rho}^{+}(L_{s}))^{3}}\right)\frac{d\bar{\rho}^{+}(L_{s})}{dL_{s}}-\bar{f}(L_{s})=:I.$

Also (1.10) yields that

$\displaystyle\left\\{1-\frac{\gamma
J^{\gamma-1}}{(\bar{u}^{+}(L_{s}))^{\gamma+1}}\right\\}\frac{d\bar{u}^{+}(L_{s})}{dL_{s}}=\left\\{1-\frac{\gamma
J^{\gamma-1}}{(\bar{u}^{-}(L_{s}))^{\gamma+1}}\right\\}\frac{d\bar{u}^{-}(L_{s})}{dL_{s}}=\frac{\bar{f}(L_{s})}{\bar{u}^{-}(L_{s})}.$

Finally, we conclude that

$\displaystyle I$ $\displaystyle=$
$\displaystyle-\left\\{\gamma(\rho^{+}(L_{s}))^{\gamma-1}-\frac{J^{2}}{(\rho^{+}(L_{s}))^{2}}\right\\}\frac{1}{\bar{u}^{+}(L_{s})}\frac{d\bar{u}^{+}(L_{s})}{dL_{s}}-\bar{f}(L_{s})$
$\displaystyle=$
$\displaystyle\frac{\bar{f}(L_{s})(\bar{u}^{+}(L_{s})-\bar{u}^{-}(L_{s}))}{\bar{u}^{-}(L_{s})}<0.$

Since the coefficients

$\displaystyle\gamma(\bar{\rho}^{+}(L_{1}))^{\gamma-2}-\frac{J^{2}}{(\bar{\rho}^{+}(L_{1}))^{3}}>0,$

then (1.11) implies that the end density $\bar{\rho}^{+}(L_{1})$ is a strictly
decreasing function of the shock position $x_{1}=L_{s}$. It follows that the
end pressure $P_{e}=(\bar{\rho}^{+}(L_{1}))^{\gamma}$ is a strictly decreasing
and continuous differentiable function on the shock position $x_{1}=L_{s}$. In
particular, when $L_{s}=L_{0}$ and $L_{s}=L_{1}$, there are two different end
pressure $P_{1},P_{2}$ with $P_{0}>P_{1}$. Hence, by the monotonicity one can
obtain a transonic shock for the end pressure $P_{e}\in(P_{1},P_{0})$. ∎

######  Remark 1.

Lemma 1.1 shows that the external force helps to stabilize the transonic shock
in flat nozzles and the shock position is uniquely determined.

The one dimensional transonic shock solution
$(\bar{u}^{\pm},\bar{\rho}^{\pm})$ with a shock occurring at $x_{1}=L_{s}$
constructed in Lemma 1.1 will be called the background solution in this paper.
The extension of the subsonic flow
$(\bar{u}^{+}(x_{1}),\bar{\rho}^{+}(x_{1}))$ of the background solution to
$L_{s}-\delta_{0}<x_{1}<L_{1}$ for a small positive number $\delta_{0}$ will
be denoted by $(\hat{u}^{+}(x_{1}),\hat{\rho}^{+}(x_{1}))$.

$x_{1}$$x_{2}$$L_{0}$$L_{1}$$x_{1}=L_{s}$$P=P_{e}+\epsilon
P_{ex}$$\quad\quad\quad\quad
supesonic$$\quad\quad\quad\quad\Omega^{-}$$\quad\quad\quad\quad
subsonic$$\quad\quad\quad\quad\Omega^{+}$$shock$$\,\,x_{1}=\xi(x_{2})$ Figure
1. Nozzle

It is natural to focus on the structural stability of this transonic shock
flows. For simplicity, we only investigate the structural stability under
suitable small perturbations of the end pressure. Therefore, the supersonic
incoming flow is unchanged and remains to be
$(\bar{u}^{-}(x_{1}),0,\bar{\rho}^{-}(x_{1}))$.

Assume that the possible shock curve $\Sigma$ and the flow behind the shock
are denoted by $x_{1}=\xi(x_{2})$ and $(u_{1}^{+},u_{2}^{+},P^{+})(x)$
respectively (See Figure 1). Let
$\Omega^{+}=\\{(x_{1},x_{2}):\xi(x_{2})<x_{1}<L_{1},-1<x_{2}<1\\}$ denotes the
subsonic region of the flow. Then the Rankine-Hugoniot conditions on $\Sigma$
gives

(1.12) $\displaystyle\begin{cases}[\rho u_{1}]-\xi^{\prime}(x_{2})[\rho
u_{2}]=0,\\\ [\rho u_{1}^{2}+P]-\xi^{\prime}(x_{2})[\rho u_{1}u_{2}]=0,\\\
[\rho u_{1}u_{2}]-\xi^{\prime}(x_{2})[\rho u_{2}^{2}+P]=0.\end{cases}$

In addition, the pressure $P$ satisfies the physical entropy conditions

(1.13) $\displaystyle P^{+}(x)>P^{-}(x)\quad\text{on}\,\Sigma.$

Since the flow is tangent to the nozzle walls $x_{2}=\pm 1$, then

(1.14) $\displaystyle u_{2}^{+}(x_{1},\pm 1)=0.$

The end pressure is perturbed by

(1.15) $\displaystyle P^{+}(L_{1},x_{2})=P_{e}+\epsilon P_{ex}(x_{2}),$

due to some technical reasons, we may readily suppose that
$P_{ex}(x_{2})=P_{e}^{\frac{1}{\gamma}}\hat{P}_{ex}(x_{2})\in
C^{2,\alpha}([-1,1])(\alpha\in(0,1))$ satisfies the compatibility conditions

(1.16) $\displaystyle\hat{P}_{ex}^{\prime}(\pm 1)=0.$

The following theorem gives the main results of this paper.

######  Theorem 1.2.

Under the assumptions on the external force and the exit pressure, there
exists a constant $\epsilon_{0}>0$ such that for all
$\epsilon\in(0,\epsilon_{0}]$, the system (1.1),(1.12)-(1.15) has a unique
transonic shock solution $(u_{1}^{+}(x),u_{2}^{+}(x),P^{+}(x);\xi(x_{2}))$
which admits the following properties:
(i). The shock $x_{1}=\xi(x_{2})\in C^{3,\alpha}([-1,1])$, and satisfies

(1.17) $\displaystyle\|\xi(x_{2})-L_{s}\|_{C^{3,\alpha}([-1,1])}\leq
C\epsilon,$

where the positive constant $C$ only depends on the background solution, the
exit pressure and $\alpha$.
(ii). The velocity and pressure in subsonic region
$(u_{1}^{+},u_{2}^{+},P^{+})(x)\in C^{2,\alpha}(\bar{\Omega}^{+})$, and there
holds

(1.18)
$\displaystyle\|(u_{1}^{+},u_{2}^{+},P^{+})(x)-(\hat{u},0,\hat{P})\|_{C^{2,\alpha}(\bar{\Omega}^{+})}\leq
C\epsilon,$

where $\Omega^{+}=\\{(x_{1},x_{2}):\xi(x_{2})<x_{1}<L_{1},-1<x_{2}<1\\}$ is
the subsonic region and
$(\hat{u},0,\hat{P})=(\hat{u}(x_{1}),0,P(\hat{\rho}(x_{1})))$ is the extended
background solution.

Our proof is influenced by the approach developed in [16, 17, 20], yet the
reformulation of the problem is different from there. It is well-known that
steady Euler equations are hyperbolic-elliptic coupled in subsonic region. The
entropy and Bernoulli’s function are conserved along the particle path, while
the pressure and the flow angle satisfy a first order elliptic system in
subsonic region. These facts are widely used in the structural stability
analysis for the transonic shock problems in flat or divergent nozzles, one
may refer to [3, 7, 8, 16, 17, 20, 22, 32, 33] and the reference therein. Here
we resort to a different decomposition based on the deformation and curl of
the velocity developed in [27, 28] for three dimensional steady Euler and
Euler-Poisson systems. The idea in that decomposition is to rewrite the
density equation as a Frobenius inner product of a symmetric matrix and the
deformation matrix by using the Bernoulli’s law. The vorticity is resolved by
an algebraic equation of the Bernoulli’s function and the entropy. We should
mention that there are several different decompositions to three dimensional
steady Euler system [2, 4, 5, 21, 23, 32] developed by many researchers for
different purposes. An interesting issue that deserves further discussion is
when using the deformation-curl decomposition to deal with the transonic shock
problem, the end pressure boundary condition becomes nonlocal since it
involves the information from the shock front (see (3.9)). However, this
nonlocal boundary condition reduces to be local after introducing the
potential function (see (3.15)).

The rest of this paper will be organized as follows. In Section 2, we
reformulate the original 2-D problem (1.1)-(1.15) by deformation-curl
decomposition developed in [27, 28] so that one can rewrite the system (1.1)
with the velocity and the Bernoulli function. We obtain a $2\times 2$ first
order system for the velocity field, a transport type equation for the
Bernoulli function and the first order ordinary differential equation for the
shock after linearization. In Section 3, we design an elaborate iteration
scheme inspired by the works [17] for the nonlinear system. The investigation
of well-posedness and regularity for the linear system are given in the
reminder part of this section. In section 4, we prove the main existence and
uniqueness theorem.

##  2\. Reformulation of the problem

Different from previous works on transonic shock problems [3, 7, 8, 16, 17,
20], we will use the deformation-curl decomposition developed in [27, 28] for
steady Euler system to decompose the original system (1.1) into an equivalent
system (2.3), where the hyperbolic quantity $B$ and elliptic quantities
$u_{1},u_{2}$ are effectively decoupled in subsonic regions. To this end,
define the Bernoulli’s function

(2.1) $\displaystyle B=\frac{1}{2}|{\bf u}|^{2}+h(\rho)-\Phi,$

where $h(s)=\frac{\gamma}{\gamma-1}s^{\gamma-1}$ is the enthalpy function.
Hence the density can be expressed by the Bernoulli function and velocity
field as

(2.2) $\displaystyle\rho=H(B,\Phi,|{\bf
u}|^{2})=\left[\frac{\gamma-1}{\gamma}(B+\Phi-\frac{1}{2}|{\bf
u}|^{2})\right]^{\frac{1}{\gamma-1}}.$

Consequently, the 2-D Euler system (1.1) with unknown function
$(u_{1},u_{2},P)$ is equivalent to the following system

(2.3)
$\displaystyle\begin{cases}\displaystyle\sum_{i,j=1}^{2}(c^{2}(H)\delta_{ij}-u_{i}u_{j})\partial_{i}u_{j}+u_{1}\partial_{1}\Phi+u_{2}\partial_{2}\Phi=0,\\\
\partial_{1}u_{2}-\partial_{2}u_{1}=-\frac{\partial_{2}B}{u_{1}},\\\
u_{1}\partial_{1}B+u_{2}\partial_{2}B=0,\end{cases}$

with unknown function $(u_{1},u_{2},B)$.

The shock curve is determined by

(2.4) $\displaystyle\xi^{\prime}(x_{2})=\frac{[\rho u_{1}u_{2}]}{[\rho
u_{2}^{2}+P]}(\xi(x_{2}),x_{2}),\quad x_{2}\in(-1,1).$

Furthermore, it follows from the R-H conditions (1.12) that

(2.5) $\displaystyle\begin{cases}[\rho u_{1}]=\frac{[\rho u_{2}][\rho
u_{1}u_{2}]}{[\rho u_{2}^{2}+P]},\\\ [\rho u_{1}^{2}+P(\rho)]=\frac{([\rho
u_{1}u_{2}])^{2}}{[\rho u_{2}^{2}+P]}.\end{cases}$

A direct computation by using (2.5) shows that on $x_{1}=\xi(x_{2})$

(2.6)
$\displaystyle(\rho^{+}(\xi(x_{2}),x_{2})-\bar{\rho}^{+}(L_{s}),u_{1}^{+}(\xi(x_{2}),x_{2})-\bar{u}^{+}(L_{s}))=$
$\displaystyle(h_{1},h_{2})(\rho^{-}(\xi(x_{2}))-\bar{\rho}^{-}(L_{s}),u^{-}(\xi(x_{2}))-\bar{u}^{-}(L_{s}),(u_{2}^{+}(\xi(x_{2}),x_{2}))^{2})$

here $h_{i}(0,0,0)=0$ for $i=1,2$. In addition, we have

(2.7) $\begin{cases}\begin{aligned} \frac{\partial
h_{1}}{\partial(\rho^{-}-\bar{\rho}^{-})}|_{(0,0,0)}&=2\bar{u}^{-}(L_{s})\frac{\bar{u}^{+}-\bar{u}^{-}}{(\bar{u}^{+}(L_{s}))^{2}-c^{2}(\bar{\rho}^{+}(L_{s}))}+1,\\\
\frac{\partial
h_{1}}{\partial(u^{-}-\bar{u}^{-})}|_{(0,0,0)}&=2\bar{\rho}^{-}(L_{s})\frac{\bar{u}^{+}-\bar{u}^{-}}{(\bar{u}^{+}(L_{s}))^{2}-c^{2}(\bar{\rho}^{+}(L_{s}))},\\\
\frac{\partial
h_{1}}{\partial(u_{2}^{+})^{2}}|_{(0,0,0)}&=\frac{(\bar{\rho}^{+}(L_{s})\bar{u}^{+}(L_{s}))^{2}}{\bar{P}^{+}(L_{s})-\bar{P}^{-}(L_{s})}\frac{1}{(\bar{u}^{+}(L_{s}))^{2}-c^{2}(\bar{\rho}^{+}(L_{s}))},\end{aligned}\end{cases}$

and

(2.8) $\begin{cases}\begin{aligned} \frac{\partial
h_{2}}{\partial(\rho^{-}-\bar{\rho}^{-})}|_{(0,0,0)}&=-(\gamma-1)\frac{\bar{P}^{+}(L_{s})-\bar{P}^{-}(L_{s})}{(\bar{\rho}^{+}(L_{s}))^{2}\bar{u}^{+}(L_{s})}\frac{\bar{u}^{+}\bar{u}^{-}}{(\bar{u}^{+}(L_{s}))^{2}-c^{2}(\bar{\rho}^{+}(L_{s}))},\\\
\frac{\partial
h_{2}}{\partial(u^{-}-\bar{u}^{-})}|_{(0,0,0)}&=\frac{2\bar{\rho}^{-}(L_{s})\bar{u}^{+}(L_{s})}{\bar{\rho}^{+}(L_{s})}\frac{\bar{u}^{-}\bar{u}^{+}}{(\bar{u}^{+}(L_{s}))^{2}-c^{2}(\bar{\rho}^{+}(L_{s}))}+\frac{\bar{\rho}^{-}(L_{s})}{\bar{\rho}^{+}(L_{s})},\\\
\frac{\partial
h_{2}}{\partial(u_{2}^{+})^{2}}|_{(0,0,0)}&=\frac{\bar{\rho}^{+}(L_{s})\bar{u}^{+}(L_{s})}{\bar{P}^{+}(L_{s})-\bar{P}^{-}(L_{s})}\frac{c^{2}(\bar{\rho}^{+}(L_{s}))}{(\bar{u}^{+}(L_{s}))^{2}-c^{2}(\bar{\rho}^{+}(L_{s}))}.\end{aligned}\end{cases}$

By substituting (2.6) into (2.1), we conclude that there is a function $h_{3}$
such that

(2.9) $\displaystyle
B^{+}(\xi(x_{2}),x_{2})-\bar{B}^{+}(L_{s})=h_{3}(\rho^{-}(\xi(x_{2}))-\bar{\rho}^{-}(L_{s}),u^{-}(\xi(x_{2}))-\bar{u}^{-}(L_{s}),(u_{2}^{+}(\xi(x_{2}),x_{2}))^{2}).$

Thus, Theorem 1.2 is established as long as we solve the problem (2.3)-(2.4)
with boundary conditions (2.5), (1.14)-(1.15). In order to deal with the free
boundary value problem (2.3)-(2.4), we introduce the following transformation
to reduce it into a fixed boundary value problem. Setting

(2.10) $\displaystyle
y_{1}=\frac{x_{1}-\xi(x_{2})}{L_{1}-\xi(x_{2})}(L_{1}-L_{s})+L_{s},\ \ \ \
y_{2}=x_{2},$

then, the domain
$\Omega^{+}=\\{(x_{1},x_{2}):\xi(x_{2})<x_{1}<L_{1},-1<x_{2}<1\\}$ is changed
into

(2.11) $\displaystyle Q=\\{(y_{1},y_{2}):L_{s}<y_{1}<L_{1},\ -1<y_{2}<1\\}.$

The inverse change variable gives

$\displaystyle
x_{1}=\xi(y_{2})+\frac{L_{1}-\xi(y_{2})}{L_{1}-L_{s}}(y_{1}-L_{s})=y_{1}+\frac{L_{1}-y_{1}}{L_{1}-L_{s}}(\xi(y_{2})-L_{s}),\
x_{2}=y_{2}.$

We now set for $y\in Q$

$\displaystyle(\tilde{u}_{j},\tilde{\rho},\tilde{B},\tilde{\Phi})(y_{1},y_{2})=(u_{j},\rho,B,\Phi)\left(\frac{L_{1}-\xi(y_{2})}{L_{1}-L_{s}}(y_{1}-L_{s})+\xi(y_{2}),y_{2}\right),\,\,j=1,2.$

The shock equation (2.4) becomes to

(2.12) $\displaystyle\xi^{\prime}(y_{2})$ $\displaystyle=\frac{(\rho
u_{1}u_{2})(\xi(y_{2}),y_{2})}{P^{+}(\rho)(\xi(y_{2}),y_{2})-P^{-}(\xi(y_{2}))+\rho(u_{2})^{2}(\xi(y_{2}),y_{2})},$
$\displaystyle=\frac{(\tilde{\rho}\tilde{u}_{1}\tilde{u}_{2})(L_{s},y_{2})}{P^{+}(\tilde{\rho})(L_{s},y_{2})-P^{-}(\xi(y_{2}))+\tilde{\rho}(\tilde{u}_{2})^{2}(L_{s},y_{2})},\,y_{2}\in(-1,1),$

and the system (2.3) is changed into

(2.13)
$\displaystyle(c^{2}(\tilde{\rho})-\tilde{u}_{1}^{2})\frac{L_{1}-L_{s}}{L_{1}-\xi(y_{2})}\partial_{y_{1}}\tilde{u}_{1}+c^{2}(\tilde{\rho})\partial_{y_{2}}\tilde{u}_{2}+\tilde{u}_{1}\frac{L_{1}-L_{s}}{L_{1}-\xi(y_{2})}\partial_{y_{1}}\tilde{\Phi}=F_{1}(\tilde{{\bf
u}},\tilde{B}),$
$\displaystyle\frac{L_{1}-L_{s}}{L_{1}-\xi(y_{2})}\partial_{y_{1}}\tilde{u}_{2}-\partial_{y_{2}}\tilde{u}_{1}-\frac{y_{1}-L_{1}}{L_{1}-\xi(y_{2})}\xi^{\prime}(y_{2})\partial_{y_{1}}\tilde{u}_{1}+\frac{\partial_{y_{2}}\tilde{B}}{\tilde{u}_{1}}=F_{2}(\tilde{\textbf{u}},\tilde{B}),$
(2.14)
$\displaystyle\tilde{u}_{1}\frac{L_{1}-L_{s}}{L_{1}-\xi(y_{2})}\partial_{y_{1}}\tilde{B}+\tilde{u}_{2}\partial_{y_{2}}\tilde{B}+\frac{y_{1}-L_{1}}{L_{1}-\xi(y_{2})}\xi^{\prime}(y_{2})\tilde{u}_{2}\partial_{y_{1}}\tilde{B}=0,$

where

$\displaystyle F_{1}(\tilde{{\bf
u}},\tilde{B})=\tilde{u}_{2}^{2}\partial_{y_{2}}\tilde{u}_{2}-(c^{2}(\tilde{\rho})-\tilde{u}_{2}^{2})\frac{y_{1}-L_{1}}{L_{1}-\xi(y_{2})}\xi^{\prime}(y_{2})\partial_{y_{1}}\tilde{u}_{2}+\tilde{u}_{1}\tilde{u}_{2}\frac{L_{1}-L_{s}}{L_{1}-\xi(y_{2})}\partial_{y_{1}}\tilde{u}_{2}$
$\displaystyle+\tilde{u}_{2}\tilde{u}_{1}(\partial_{y_{2}}\tilde{u}_{1}+\frac{y_{1}-L_{1}}{L_{1}-\xi(y_{2})}\xi^{\prime}(y_{2})\partial_{y_{1}}\tilde{u}_{1})-\tilde{u}_{2}(\partial_{y_{2}}\tilde{\Phi}+\frac{y_{1}-L_{1}}{L_{1}-\xi(y_{2})}\xi^{\prime}(y_{2})\partial_{y_{1}}\tilde{\Phi}),$
$\displaystyle F_{2}(\tilde{{\bf
u}},\tilde{B})=-{\frac{y_{1}-L_{1}}{L_{1}-\xi(y_{2})}\frac{\xi^{\prime}(y_{2})}{\tilde{u}_{1}}\partial_{y_{1}}\tilde{B}}.$

Consider the perturbed functions $v_{i}(y_{1},y_{2})$, $i=1,2,3,4,$ as

$\displaystyle
v_{1}(y_{1},y_{2})=\tilde{u}_{1}(y_{1},y_{2})-\bar{u}^{+}(y_{1}),\ \
v_{2}(y_{1},y_{2})=\tilde{u}_{2}(y_{1},y_{2}),$ $\displaystyle
v_{3}(y_{1},y_{2})=\tilde{B}(y_{1},y_{2})-\bar{B}^{+},\ \ \
v_{4}(y_{2})=\xi(y_{2})-L_{s},$

and define the vector functions

(2.15) $\displaystyle
V(y_{1},y_{2})=(v_{1}(y_{1},y_{2}),v_{2}(y_{1},y_{2}),v_{3}(y_{1},y_{2}),v_{4}(y_{2})).$

It follows from (2.12) that the shock satisfies

(2.16) $\displaystyle
v_{4}^{\prime}(y_{2})=\frac{\tilde{\rho}(\bar{u}+v_{1})v_{2}(L_{s},y_{2})}{P^{+}(\tilde{\rho})(L_{s},y_{2})-P^{-}(\xi(y_{2}))+\tilde{\rho}(\tilde{u}_{2})^{2}(L_{s},y_{2})}.$

Through a direct computation, one can derive from (2.9) and (2.14) that the
Bernoulli function satisfy a transport type equation

(2.17)
$\displaystyle\begin{cases}[(\bar{u}^{+}+v_{1})(L_{1}-L_{s})+v_{2}(y_{1}-L_{1})v_{4}^{\prime}(y_{2})]\partial_{y_{1}}v_{3}+v_{2}(L_{1}-v_{4}-L_{s})\partial_{y_{2}}v_{3}=0,\\\
v_{3}(L_{s},y_{2})=b_{3}v_{4}(y_{2})+R_{3}(y_{2}),\end{cases}$

where

(2.18) $\displaystyle
b_{3}=\frac{\bar{\rho}^{-}(L_{s})-\bar{\rho}^{+}(L_{s})}{\bar{\rho}^{+}(L_{s})}\bar{f}(L_{s}),$

and $R_{3}(y_{2})=R_{3}(V(L_{s},y_{2}))=O(|V(L_{s},y_{2})|)^{2}$ is an error
term of second order. We may readily drop superscribe $+$ on the background
solutions if there is no risk of confusing. And the first order system for
$v_{1},v_{2}$ is given by,

(2.19)
$\displaystyle\begin{cases}(c^{2}(\bar{\rho}^{+})-(\bar{u}^{+})^{2})\partial_{y_{1}}v_{1}+c^{2}(\bar{\rho}^{+})\partial_{y_{2}}v_{2}+B_{1}(y_{1})v_{1}\\\
\,\,\quad\quad\quad+B_{3}(y_{1})v_{3}+B_{4}(y_{1})v_{4}(y_{2})=F_{3}(V,\nabla
V),\\\
\partial_{y_{1}}v_{2}-\partial_{y_{2}}v_{1}+\frac{L_{1}-y_{1}}{L_{1}-L_{s}}\bar{u}^{\prime}v_{4}^{\prime}+\frac{\partial_{y_{2}}v_{3}}{\bar{u}}=F_{4}(V,\nabla
V),\end{cases}$

where $F_{3},F_{4}$ represent the remainder term of second order with respect
to $V$ and $\nabla V$, and

$\displaystyle
B_{1}(y_{1})=\bar{f}(y_{1})-(\gamma+1)\bar{u}\bar{u}^{\prime}=\frac{\gamma\bar{u}^{2}+c^{2}(\bar{\rho}^{+})}{c^{2}(\bar{\rho}^{+})-\bar{u}^{2}}\bar{f}>0,\,$
$\displaystyle B_{3}(y_{1})=(\gamma-1)\bar{u}^{\prime},\,$ $\displaystyle
B_{4}(y_{1})=\frac{1}{L_{1}-L_{s}}[(\gamma-1)\bar{f}(y_{1})(L_{1}-y_{1})\bar{u}^{\prime}-\bar{u}\bar{f}+\bar{u}\bar{f}^{\prime}(y_{1})(L_{1}-y_{1})].$

It’s obvious that the second formula in (2.6) gives the boundary condition of
$v_{1}$ on the entrance $x_{1}=L_{s}$. Meanwhile, the formula (2.2) after
changing variable becomes

(2.20) $\displaystyle c^{2}(\tilde{\rho})(y)$
$\displaystyle=\gamma\tilde{\rho}^{\gamma-1}=(\gamma-1)(\tilde{B}-\frac{1}{2}|\tilde{\bf
u}|^{2}+\bar{\Phi})$
$\displaystyle=(\gamma-1)(\bar{B}^{+}+v_{3}-\frac{1}{2}(\bar{u}^{+}+v_{1})^{2}-\frac{1}{2}v_{2}^{2}+\bar{\Phi})$
$\displaystyle=c^{2}(\bar{\rho}^{+})+(\gamma-1)(v_{3}-\bar{u}^{+}v_{1}-\frac{1}{2}v_{1}^{2}-\frac{1}{2}v_{2}^{2}),$

which together with (1.15) gives the boundary condition of $v_{1}$ on the exit
$x_{1}=L_{1}$. Hence, the boundary conditions to the system (2.19) read as
follow

(2.21)
$\displaystyle\begin{cases}v_{1}(L_{s},y_{2})=b_{2}v_{4}(y_{2})+R_{2}(y_{2}),\\\
v_{1}(L_{1},y_{2})=\frac{1}{\bar{u}(L_{1})}(v_{3}(L_{1},y_{2})-\epsilon\hat{P}_{ex}(y_{2}))+R_{4}(y_{2}),\\\
v_{2}(y_{1},\pm 1)=0,\end{cases}$

here

(2.22) $\displaystyle
b_{2}=\frac{\bar{\rho}^{-}(L_{s})\bar{u}^{+}(L_{s})}{\bar{\rho}^{+}(L_{s})[(\bar{u}^{+}(L_{s}))^{2}-c^{2}(\bar{\rho}^{+}(L_{s}))]}\bar{f}(L_{s})<0,$

and $R_{2}(y_{2})=R_{2}(V(L_{s},y_{2}))=O(|V(L_{s},y_{2})|)^{2}$,
$R_{4}(y_{2})=R_{4}(V(L_{1},y_{2}))=O(|V(L_{1},y_{2})|)^{2}$ are error terms
of second order. Based on our reformulation, Theorem 1.2 follows from the
following results.

######  Theorem 2.1.

Under the same assumptions as in Theorem 1.2 there exists a positive constant
$\epsilon_{0}>0$ such that for each $\epsilon\in(0,\epsilon_{0}]$, the system
(2.16)-(2.21) has a unique solution $V\in(C^{2,\alpha}(\bar{Q}))^{3}\times
C^{3,\alpha}([-1,1])$ satisfying the following estimate

(2.23)
$\displaystyle\sum_{i=1}^{3}\|v_{i}\|_{C^{2,\alpha}(\bar{Q})}+\|v_{4}\|_{C^{3,\alpha}([-1,1])}\leq
C\epsilon,$

where the constant $C$ depends only on the background solution, the exit
pressure and $\alpha\in(0,1)$.

##  3\. iteration scheme and the linear system

In the first part of this section, we construct an iteration scheme for the
nonlinear system, and the problem is reduced to the solvability of
corresponding linear systems. Indeed, it turns out that the linear system is a
non-local elliptic equation of second order with a free parameter denoting the
relative location of the shock position on the wall $x_{2}=-1$. Then we study
the existence, uniqueness and regularity for this linear system in the
remainder part of this section.

###  3.1. Iteration Scheme

Inspired by [17], we will develop an iteration scheme to prove Theroem 2.1.
Consider the Banach space

(3.1)
$\displaystyle{\mathcal{V}}_{\delta}:=\\{V:\displaystyle\sum_{i=1}^{3}\|v_{i}\|_{C^{2,\alpha}(\bar{Q})}+|v_{4}|_{C^{2,\alpha}[-1,1]}\leq\delta;\partial_{y_{2}}v_{j}(y_{1},\pm
1)=0,$ $\displaystyle j=1,3;v_{2}(y_{1},\pm
1)=\partial_{y_{2}}^{2}v_{2}(y_{1},\pm 1)=0;v_{4}^{\prime}(\pm
1)=v_{4}^{(3)}(\pm 1)=0\\},$

here $\delta>0$ is a small constant to be determined later. For a fix
$\hat{V}\in{\mathcal{V}}_{\delta}$, equivalently, we have the following
quantity

$\displaystyle(\hat{v}_{1},\hat{v}_{2},\hat{B},\hat{\rho},\hat{P},\hat{\xi})(y).$

We now define the linearized scheme to the problem (2.16)-(2.21) as follows.

Firstly, thanks to (2.16), $v_{4}$ is determined by

(3.2) $\displaystyle
v_{4}^{\prime}(y_{2})=b_{0}v_{2}(L_{s},y_{2})+F_{5}(\hat{V})(L_{s},y_{2}),$

where

$\displaystyle
b_{0}=\frac{(\bar{\rho}\bar{u})(L_{s})}{P^{+}(\bar{\rho})(L_{s})-P^{-}(L_{s})}>0,$
$\displaystyle
F_{5}(y_{2})=\\{\frac{\hat{\rho}\hat{u}_{1}}{P^{+}(\hat{\rho})(L_{s},y_{2})-P^{-}(\hat{\xi}(y_{2}))+\hat{\rho}(\hat{u}_{2})^{2}(L_{s},y_{2})}-b_{0}\\}v_{2}(L_{s},y_{2}),$

hence, one can express the shock as

(3.3) $\displaystyle
v_{4}(y_{2})=v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau+R_{5}(y_{2}),$

where $R_{5}(y_{2})=\int_{-1}^{y_{2}}F_{5}(\tau)d\tau$ is a error term of
second order. Due to $\hat{V}\in{\mathcal{V}}_{\delta}$, we have

(3.4) $\displaystyle F_{5}(\pm 1)=F_{5}^{{}^{\prime\prime}}(\pm
1)=0,\,\|F_{5}\|_{C^{k,\alpha}[-1,1]}\leq
C\delta\|\hat{v}_{2}\|_{C^{k,\alpha}(\bar{Q})},\,k=0,1,2.$

Secondly, Using (2.17), we get the linearized transport equation for $v_{3}$:

$\displaystyle[(\bar{u}+\hat{v}_{1})(L_{1}-L_{s})+\hat{v}_{2}(y_{1}-L_{1})\hat{v}_{4}^{\prime}(y_{2})]\partial_{y_{1}}v_{3}+\hat{v}_{2}(L_{1}-\hat{v}_{4}-L_{s})\partial_{y_{2}}v_{3}=0\,\,\text{in}\,Q,$

with initial data

$\displaystyle v_{3}(L_{s},y_{2})=b_{3}v_{4}(y_{2})+R_{3}(y_{2}).$

Thus, it can be solved by characteristic methods. Let $y_{2}(s;\beta)$ be the
characteristics going through $(y_{1},y_{2})$ with $y_{2}(L_{s})=\beta$, i.e.

(3.5)
$\displaystyle\begin{cases}\frac{dy_{2}}{ds}(s;\beta)=\frac{\hat{v}_{2}(L_{1}-\hat{v}_{4}-L_{s})}{(\bar{u}+\hat{v}_{1})(L_{1}-L_{s})+\hat{v}_{2}(y_{1}-L_{1})\hat{v}_{4}^{\prime}(y_{2})},\
\ L_{s}<s<L_{1},\\\ y_{2}(L_{s})=\beta.\end{cases}$

It is noted that $\beta$ can be also regarded as a function of
$y=(y_{1},y_{2})$, this is denoted by $\beta=\beta(y)$, which leads to

(3.6) $\displaystyle
v_{3}(y_{1},y_{2})=v_{3}(L_{s},\beta(y))=b_{3}v_{4}(y_{2})+F_{6}(y),$

where

$\displaystyle
F_{6}(y)=b_{3}\int_{y_{2}}^{\beta(y)}v_{4}^{\prime}(\tau)d\tau+R_{3}(\hat{V}(L_{s},\beta(y)))$

is an error term of second order. Furthermore, we have

(3.7) $\displaystyle\partial_{y_{2}}F_{6}(y_{1},\pm 1)=0,$
$\displaystyle\|F_{6}\|_{C^{k,\alpha}(\bar{Q})}\leq
C\delta(\sum_{i=1}^{3}\|\hat{v}_{i}\|_{C^{k,\alpha}(\bar{Q})}+\|\hat{v}_{4}\|_{C^{k+1,\alpha}(\bar{Q})}),\,k=0,1,2.$

It remains to determine the velocity $v_{1},v_{2}$ and the shock position
difference $v_{4}(-1)$ on the wall $x_{2}=-1$. Substituting (3.3) and (3.6)
into (2.19) and (2.21), we get the following linearized system for
$v_{1},v_{2}$ with a unknown parameter $v_{4}(-1)$:

(3.8)
$\displaystyle\begin{cases}\partial_{y_{1}}v_{1}+\frac{1}{1-\bar{M}^{2}}\partial_{y_{2}}v_{2}+\frac{B_{1}(y_{1})}{c^{2}(\bar{\rho}^{+})-(\bar{u}^{+})^{2}}v_{1}\\\
\quad\quad+\frac{B_{3}(y_{1})b_{3}+B_{4}(y_{1})}{c^{2}(\bar{\rho}^{+})-(\bar{u}^{+})^{2}}(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau)=G_{1}(y),\\\
\partial_{y_{1}}v_{2}-\partial_{y_{2}}[v_{1}-\lambda(y_{1})(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau)]=G_{2}(y),\end{cases}$

and the boundary conditions

(3.9)
$\displaystyle\begin{cases}v_{1}(L_{s},y_{2})=b_{2}(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau)+R_{6}(y_{2}),\\\
v_{1}(L_{1},y_{2})=\frac{b_{3}}{\bar{u}(L_{1})}(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau)-\frac{\epsilon\hat{P}_{ex}(y_{2})}{\bar{u}(L_{1})}+R_{7}(y_{2}),\\\
v_{2}(y_{1},\pm 1)=0,\end{cases}$

where

$\displaystyle\lambda(y_{1})=\frac{L_{1}-y_{1}}{L_{1}-L_{s}}\bar{u}^{\prime}+\frac{b_{3}}{\bar{u}},$
$\displaystyle R_{6}(y_{2})=b_{2}R_{5}(y_{2})+R_{2}(y_{2}),$ $\displaystyle
R_{7}(y_{2})=\frac{b_{3}R_{5}(y_{2})+F_{6}(L_{1},y_{2})}{\bar{u}(L_{1})}+R_{4}(y_{2}),$
$\displaystyle
G_{1}(y)=\frac{F_{3}(\hat{V},\nabla\hat{V})-B_{3}(y_{1})R_{5}(y_{2})-B_{3}(y_{1})F_{6}(y)-B_{4}(y_{1})R_{5}(y_{2})}{(c^{2}(\bar{\rho}^{+})-(\bar{u}^{+})^{2})},$
$\displaystyle
G_{2}(y)=F_{4}(\hat{V},\nabla\hat{V})-\frac{\partial_{y_{2}}F_{6}(y)}{\bar{u}}+\lambda(y_{1})\partial_{y_{2}}R_{5}(y_{2}),$

are error terms of second order. It follows from (3.4),(3.7) and a simple
calculation that

(3.10) $\displaystyle\partial_{y_{2}}G_{1}(y_{1},\pm 1)=0,\,G_{2}(y_{1},\pm
1)=0,$ $\displaystyle\|G_{i}\|_{C^{k-1,\alpha}(\bar{Q})}\leq
C\delta\|\hat{V}\|_{C^{k,\alpha}(\bar{Q})},\,i,k=1,2,$

and

(3.11) $\displaystyle R_{6}^{\prime}(\pm 1)=0,\,R_{7}^{\prime}(\pm 1)=0,$
$\displaystyle\|(R_{6},R_{7})\|_{C^{k,\alpha}[-1,1]}\leq
C\delta\|\hat{V}\|_{C^{k,\alpha}(\bar{Q})},\,k=0,1,2.$

The second equation in (3.8) implies that there is a potential function
$\phi(y)$ satisfy

(3.12)
$\displaystyle\begin{cases}\partial_{y_{2}}\phi=v_{2},\,\,\phi(L_{s},-1)=0,\\\
\partial_{y_{1}}\phi=v_{1}-\lambda(y_{1})(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau)+\int_{-1}^{y_{2}}G_{2}(y_{1},\tau)d\tau,\end{cases}$

it follows that $v_{1},v_{2}$ can be represented by

(3.13) $\displaystyle\begin{cases}v_{2}=\partial_{y_{2}}\phi,\\\
v_{1}=\partial_{y_{1}}\phi+\lambda(y_{1})[v_{4}(-1)+b_{0}\phi(L_{s},y_{2})]-\int_{-1}^{y_{2}}G_{2}(y_{1},\tau)d\tau.\end{cases}$

Substituting (3.13) into the first equation in (3.8), we conclude that $\phi$
satisfying the following non-local elliptic equation of second with an unknown
constat $v_{4}(-1)$

(3.14)
$\displaystyle\partial_{y_{1}}^{2}\phi+\frac{1}{1-\bar{M}^{2}}\partial_{y_{2}}^{2}\phi+\lambda_{1}(y_{1})\partial_{y_{1}}\phi+\lambda_{0}(y_{1})b_{0}(\frac{v_{4}(-1)}{b_{0}}+\phi(L_{s},y_{2}))$
$\displaystyle=G_{1}(y)+\lambda_{1}(y_{1})\int_{-1}^{y_{2}}G_{2}(y_{1},\tau)d\tau+\partial_{y_{1}}\int_{-1}^{y_{2}}G_{2}(y_{1},\tau)d\tau,$

where

$\displaystyle\lambda_{1}(y_{1})=\frac{B_{1}(y_{1})}{c^{2}(\bar{\rho})-\bar{u}^{2}}>0,$
$\displaystyle\lambda_{0}(y_{1})=\frac{1}{c^{2}(\bar{\rho})-\bar{u}^{2}}[(c^{2}(\bar{\rho})-\bar{u}^{2})\lambda^{\prime}+B_{1}\lambda+B_{3}b_{3}+B_{4}].$

Similarly, substituting (3.13) into the boundary conditions (3.9), combine
with the boundary condition of $\phi$ in (3.12), we have

(3.15)
$\displaystyle\begin{cases}\partial_{y_{1}}\phi(L_{s},y_{2})=b_{0}(b_{2}-\lambda(L_{s}))(\frac{v_{4}(-1)}{b_{0}}+\phi(L_{s},y_{2}))+R_{6}(y_{2})+\int_{-1}^{y_{2}}G_{2}(L_{s},\tau)d\tau,\\\
\partial_{y_{1}}\phi(L_{1},y_{2})=-\frac{\epsilon\hat{P}_{ex}(y_{2})}{\bar{u}(L_{1})}+R_{7}(y_{2})+\int_{-1}^{y_{2}}G_{2}(L_{1},\tau)d\tau,\\\
\partial_{y_{2}}\phi(y_{1},\pm 1)=\phi(L_{s},-1)=0.\end{cases}$

So far, we have reduced the problem (3.8)-(3.9) into a non-local elliptic
equation of $\phi$ with an unknown constant $v_{4}(-1)$. Hence, it is
sufficient to establish the solvability and regularity results for the problem
(3.14)-(3.15) in order to study the original problem. We are going to do it in
the next subsection.

###  3.2. A non-local elliptic equation with a free constant

In this section, we prove the existence, uniqueness and regularity results for
the problem (3.14). To this end, we consider the following more concise form
of second order elliptic system with an unknown constant $\kappa$

(3.16)
$\displaystyle\begin{cases}\partial_{y_{1}}^{2}\phi+a_{2}(y_{1})\partial_{y_{2}}^{2}\phi+a_{1}(y_{1})\partial_{y_{1}}\phi-
a_{0}(y_{1})(\kappa+\phi(L_{s},y_{2}))=\partial_{y_{1}}f,\,\text{in}\,Q,\\\
\partial_{y_{1}}\phi(L_{s},y_{2})-a_{3}(\kappa+\phi(L_{s},y_{2}))=g_{1}(y_{2}),\\\
\partial_{y_{1}}\phi(L_{1},y_{2})=g_{2}(y_{2}),\\\
\partial_{y_{2}}\phi(y_{1},\pm 1)=\phi(L_{s},-1)=0,\\\ \end{cases}$

where the smooth coefficients $a_{i}(y_{1})$, i=0,1,2 and the constant $a_{3}$
satisfy

(3.17) $\displaystyle a_{1}(y_{1})<C_{1},\,0<C_{0}<a_{i}(y)<C_{1},,i=0,2,3,\,$

and the parameter $\kappa$ is a constant to be determined with the solution
itself.

The first lemma implies that the inhomogeneous problem corresponding to the
system (3.16) without the unknown constant has a unique weak solution.

######  Lemma 3.1.

Suppose that $f\in L^{2}(Q)$ and $g_{i}\in L^{2}(-1,1),i=1,2$, then, there
exists a suitable large positive constant $K$, such that the following
inhomogeneous second order elliptic equation

(3.18) $\displaystyle\begin{cases}-\partial_{y_{1}}^{2}\phi-
a_{2}(y_{1})\partial_{y_{2}}^{2}\phi-
a_{1}(y_{1})\partial_{y_{1}}\phi+K\phi+a_{0}(y_{1})\phi(L_{s},y_{2})=\partial_{y_{1}}f,\,\text{in}\,Q,\\\
\partial_{y_{1}}\phi(L_{s},y_{2})-a_{3}\phi(L_{s},y_{2})=g_{1}(y_{2}),\\\
\partial_{y_{1}}\phi(L_{1},y_{2})=g_{2}(y_{2}),\\\
\partial_{y_{2}}\phi(y_{1},\pm 1)=0,\\\ \end{cases}$

admits a unique weak solution $\phi\in H^{1}(Q)$ satisfying

(3.19) $\displaystyle\|\phi\|_{H^{1}(Q)}\leq
C(\|(g_{1},g_{2})\|_{L^{2}(-1,1)}+\|f\|_{L^{2}(Q)}),$

for some positive constant $C>0$.

###### Proof.

For $\phi,\psi\in H^{1}(Q)$, define the bilinear form

$\displaystyle{\mathcal{B}}[\phi,\psi]=\int_{Q}\partial_{y_{1}}\phi\partial_{y_{2}}\psi
dy+\int_{Q}a_{2}(y_{1})\partial_{y_{2}}\phi\partial_{y_{2}}\psi
dy-\int_{Q}a_{1}(y_{1})\psi\partial_{y_{1}}\phi dy$
$\displaystyle\quad\quad+K\int_{Q}\phi\psi
dy+\int_{Q}a_{0}(y_{1})\phi(L_{s},y_{2})\psi
dy+a_{3}\int_{-1}^{1}\phi(L_{s},y_{2})\psi(L_{s},y_{2})dy_{2},$

and the linear functional on $H^{1}(Q)$

$\displaystyle
l(\psi)=\int_{-1}^{1}g_{2}(y_{2})\psi(L_{1},y_{2})dy_{2}-\int_{-1}^{1}g_{1}(y_{2})\psi(L_{s},y_{2})dy_{2}-\int_{Q}\partial_{y_{1}}f\psi
dy.$

It’s obviously that the linear functional $l(\psi)$ on $H^{1}(Q)$ is
continuous, i.e,

(3.20) $\displaystyle|l(\psi)|\leq
C(\|(g_{1},g_{2})\|_{L^{2}(-1,1)}+\|f\|_{L^{2}(Q)})\|\psi\|_{H^{1}(Q)},$

where we have used the trace theorem. So, what we need to do is just verify
that the conditions of Lax-Milgram Theorem is satisfied for the bilinear form
${\mathcal{B}}$. The boundedness of ${\mathcal{B}}_{K}$ is trivial, we will
show that ${\mathcal{B}}_{K}$ is also coercive. Denote
$\Lambda=\min\\{1,C_{0}\\}>0$, then

$\displaystyle\Lambda\int_{Q}|\nabla\phi|^{2}dy+K\int_{Q}|\phi|^{2}dy\leq{\mathcal{B}}[\phi,\phi]+\int_{Q}a_{1}(y_{1})\phi\partial_{y_{1}}\phi
dy$ $\displaystyle-\int_{Q}a_{0}(y_{1})\phi(L_{s},y_{2})\phi(y_{1},y_{2})dy-
a_{3}\int_{-1}^{1}|\phi(L_{s},y_{2})|^{2}dy_{2},$

Cauchy’s inequality gives

$\displaystyle\int_{Q}a_{1}(y_{1})\phi\partial_{y_{1}}\phi dy\leq
C_{1}\epsilon\int_{Q}|\nabla\phi|^{2}dy+\frac{C_{1}}{4\epsilon}\int_{Q}|\phi|^{2}dy,$

and

$\displaystyle\int_{Q}a_{0}(y_{1})\phi(L_{s},y_{2})\phi(y_{1},y_{2})dy\leq
C_{tr}(L_{1}-L_{s})C_{1}\epsilon\int_{Q}|\nabla\phi|^{2}dy+\frac{C_{1}}{4\epsilon}\int_{Q}|\phi|^{2}dy,$

Then, fix $\epsilon_{0}$ such that
$C_{1}\epsilon_{0}(1+(L_{1}-L_{s})C_{tr})<\frac{\Lambda}{2}$, and choosing
$K=\max\\{\Lambda,\frac{C_{1}}{\epsilon_{0}}\\}$, thanks to the positivity of
$a_{3}$, we obtain

$\displaystyle{\mathcal{B}}[\phi,\phi]\geq\frac{\Lambda}{2}\|\phi\|_{H^{1}(Q)},$

the unique existence follows from the Lax-Milgram Theorem and (3.20) gives the
estimates (3.19). Thus the proof is completed. ∎

The unique existence of regular solution to the non-local system (3.16) is
established in the following proposition.

######  Proposition 3.2.

For any $f\in C^{1,\alpha}(\bar{Q})$, $g_{i}\in C^{\alpha}(\bar{Q})$, there is
a unique weak solution $(\phi,\kappa)$, such that $\phi\in H^{1}(Q)$ and the
following estimate holds

(3.21) $\displaystyle\|\phi\|_{H^{1}(Q)}+|\kappa|\leq
C(\|f\|_{C^{\alpha}(\bar{Q})}+|(g_{1},g_{2})|_{C^{1,\alpha}[-1,1]}).$

Moreover, if the compatibility conditions

(3.22)
$\displaystyle\partial_{y_{2}}f(y_{1},-1)=\partial_{y_{2}}f(y_{1},1)=0,\,g_{i}^{\prime}(-1)=g_{i}^{\prime}(1)=0,\,i=1,2,$

are fulfilled, then $\phi\in C^{2,\alpha}(\bar{Q})$

(3.23) $\displaystyle\|\phi\|_{C^{1,\alpha}(\bar{Q})}\leq
C(\|f\|_{C^{\alpha}(\bar{Q})}+|(g_{1},g_{2})|_{C^{\alpha}[-1,1]}+\|\phi\|_{H^{1}(Q)}+|\kappa|),$

and

(3.24) $\displaystyle\|\phi\|_{C^{2,\alpha}(\bar{Q})}\leq
C(\|f\|_{C^{1,\alpha}(\bar{Q})}+|(g_{1},g_{2})|_{C^{1,\alpha}[-1,1]}+\|\phi\|_{H^{1}(Q)}+|\kappa|).$

for some positive constant $C>0$.

###### Proof.

The proof is divided into two steps.
Step 1: Regularity of weak solutions. we will use the symmetric extension
methods to exclude the possible singularities that may appear at the corner,
which implies that the weak solution $\phi\in H^{1}(Q)$ to the system (3.16)
are essentially more regular. To this end, introduce the notation

$\displaystyle Q^{*}:=\\{(y_{1},y_{2}):L_{s}<y_{1}<L_{1},\ -2<y_{2}<2\\}.$

and define the extended function $\phi^{*}(y)$ on $Q^{*}$ as

(3.25)
$\displaystyle\phi^{*}(y)=\begin{cases}\phi(y_{1},2-y_{2}),\,1<y_{2}<2,\\\
\phi(y_{1},y_{2}),\,-1<y_{2}<1,\\\
\phi(y_{1},-2-y_{2}),-2<y_{2}<-1.\end{cases}$

Then the extended function $\phi^{*}$ satisfies

(3.26)
$\displaystyle\begin{cases}\partial_{y_{1}}^{2}\phi^{*}+a_{2}(y_{1})\partial_{y_{2}}^{2}\phi^{*}+a_{1}(y_{1})\partial_{y_{1}}\phi^{*}-a_{0}(y_{1})(\kappa+\phi^{*}(L_{s},y_{2}))=\partial_{y_{1}}f^{*},\,\text{in}\,Q^{*},\\\
\partial_{y_{1}}\phi^{*}(L_{s},y_{2})-a_{3}(\kappa+\phi^{*}(L_{s},y_{2}))=g^{*}_{1}(y_{2}),\\\
\partial_{y_{1}}\phi^{*}(L_{1},y_{2})=g^{*}_{2}(y_{2}),\\\
\partial_{y_{2}}\phi^{*}(y_{1},\pm 1)=\phi^{*}(L_{s},-1)=0,\\\ \end{cases}$

Using the standard interior and the boundary estimates for the second order
linear elliptic equations in [13], we obtain that $\phi^{*}(y)\in
C^{1,\alpha}(Q^{*})$ and

(3.27) $\displaystyle\|\phi^{*}\|_{C^{1,\alpha}(Q^{*})}\leq
C(\|f^{*}\|_{C^{1,\alpha}(Q^{*})}+\|(g_{1},g_{2})\|_{C^{1,\alpha}[-2,2]}+\|\phi^{*}(L_{s},y_{2})\|_{L^{2}[-2,2]}+|\kappa|),$

which implies that $\phi^{*}(L_{s},y_{2})\in C^{1,\alpha}[-2,2]$. Use the
estimate (3.27) again to conclude that $\phi^{*}(y)\in C^{2,\alpha}(Q^{*})$
and

(3.28) $\displaystyle\|\phi^{*}\|_{C^{1,\alpha}(Q^{*})}\leq
C(\|f^{*}\|_{C^{1,\alpha}(Q^{*})}+\|(g_{1},g_{2})\|_{C^{1,\alpha}[-2,2]}+\|\phi^{*}(0,y_{2})\|_{L^{2}[-2,2]}+|\kappa|).$

Then (3.23) and (3.24) follows immediately.
Step 2: Existence and Uniqueness of weak solutions. Due to the linearity, any
solution $\phi$ to the problem (3.16) can be decomposed as
$\phi=\phi_{1}+\phi_{2}$, where $\phi_{i}$, i=1,2, satisfy the following
equation respectively

(3.29)
$\displaystyle\begin{cases}\partial_{y_{1}}^{2}\phi_{1}+a_{2}(y_{1})\partial_{y_{2}}^{2}\phi_{1}+a_{1}(y_{1})\partial_{y_{1}}\phi_{1}-a_{0}(y_{1})\phi_{1}(L_{s},y_{2})=\partial_{y_{1}}f,\,\text{in}\,Q,\\\
\partial_{y_{1}}\phi_{1}(L_{s},y_{2})-a_{3}\phi_{1}(L_{s},y_{2})=g_{1}(y_{2}),\\\
\partial_{y_{1}}\phi_{1}(L_{1},y_{2})=g_{2}(y_{2}),\\\
\partial_{y_{2}}\phi_{1}(y_{1},\pm 1)=0,\end{cases}$

and

(3.30)
$\displaystyle\begin{cases}\partial_{y_{1}}^{2}\phi_{2}+a_{2}(y_{1})\partial_{y_{2}}^{2}\phi_{2}+a_{1}(y_{1})\partial_{y_{1}}\phi_{2}-a_{0}(y_{1})(\kappa+\phi_{2}(L_{s},y_{2}))=0,\,\text{in}\,Q,\\\
\partial_{y_{1}}\phi_{2}(L_{s},y_{2})-a_{3}(\kappa+\phi_{2}(L_{s},y_{2}))=0,\\\
\partial_{y_{1}}\phi_{2}(L_{1},y_{2})=\partial_{y_{2}}\phi_{2}(y_{1},\pm
1)=0,\\\ \phi_{2}(L_{s},-1)=-\phi_{1}(L_{s},-1).\end{cases}$

Combing the Lemma 3.1 with the Fredholm alternative theorem, one can easily
derive (3.29) has a unique $H^{1}(Q)$ solution $\phi_{1}$ which satisfying
(3.21). On the other hand, one can prove that the only weak solution to (3.30)
must be $(\phi_{2},\kappa)=(-\phi_{1}(L_{s},-1),\phi_{1}(L_{s},-1))$ by
applying the maximum principle. For the detailed proof, one can refer to Lemma
4.1 in [17]. Thus the proof is completed. ∎

At this point, we can easily illustrate the well-posedness to the reformulated
problem (3.14)-(3.15).

######  Lemma 3.3.

The problem (3.14)-(3.15) has a unique solution $(\phi,\kappa)\in
C^{2,\alpha}(\bar{Q})\times\mathbb{R}$ satisfying

(3.31) $\displaystyle\|\phi\|_{C^{k,\alpha}(\bar{Q})}+|\kappa|\leq
C(\|(G_{1},G_{2})\|_{C^{k-1,\alpha}(\bar{Q})}$
$\displaystyle+\|(R_{6}(y_{2}),R_{7}(y_{2}))\|_{{}_{C^{1,\alpha}[-1,1]}}+\|\epsilon\hat{P}_{ex}(y_{2})\|_{C^{1,\alpha}[-1,1]}),\,k=1,2,$

for some positive constant C.

###### Proof.

It suffices to verify the solvability condition (3.17) for the problem
(3.14)-(3.15). A direct but tedious computation shows that

$\displaystyle
a_{0}(y_{1})=-b_{0}\lambda_{0}(y_{1})=-\frac{2c^{2}(\bar{\rho}^{+})\bar{f}b_{0}b_{3}}{\bar{u}(c^{2}(\bar{\rho}^{+})-\bar{u}^{2})^{2}}>0,$
$\displaystyle
a_{1}(y_{1})=\lambda_{1}(y_{1})=\frac{\gamma\bar{u}^{2}+c^{2}(\bar{\rho}^{+})}{(c^{2}(\bar{\rho}^{+})-\bar{u}^{2})^{2}}\bar{f}>0,$
$\displaystyle a_{2}(y_{1})=\frac{1}{1-\bar{M}^{2}}>0,$ $\displaystyle
a_{3}=b_{0}(b_{2}-\lambda(L_{s}))=b_{0}\frac{c^{2}(\bar{\rho}^{+})(\bar{\rho}^{+}-\bar{\rho}^{-})\bar{f}(L_{s})}{\bar{\rho}^{+}\bar{u}(c^{2}(\bar{\rho}^{+})-\bar{u}^{2})}>0,$

since the background solution is subsonic and smooth, the upper bound is
trivial. Hence, Proposition 3.2 implies that there exist a unique weak
solution $(\phi,\kappa)$, and the estimate (3.31) follows from (3.23) and
(3.24). ∎

In view of the analysis of the problem (3.14)-(3.15), the well-posedness of
the equation (3.8)-(3.9) follows.

######  Lemma 3.4.

The problem (3.8)-(3.9) admits a unique solution $(v_{1},v_{2},v_{4}(-1))\in
C^{2,\alpha}(\bar{Q})\times\mathbb{R}$ satisfying

(3.32) $\displaystyle\|(v_{1},v_{2})\|_{C^{k,\alpha}(\bar{Q})}+|v_{4}(-1)|\leq
C(\|(G_{1},G_{2})\|_{C^{k-1,\alpha}(\bar{Q})}$
$\displaystyle+\|(R_{6}(y_{2}),R_{7}(y_{2}))\|_{{}_{C^{k,\alpha}[-1,1]}}+\|\epsilon\hat{P}_{ex}(y_{2})\|_{C^{k,\alpha}[-1,1]}),\,\,k=1,2,$

and the compatibility conditions

(3.33) $\displaystyle\partial_{y_{2}}v_{1}(y_{1},\pm
1)=0,\,\,\partial_{y_{2}}v_{2}(y_{1},\pm
1)=\partial^{2}_{y_{2}}v_{2}(y_{1},\pm 1)=0.$

###### Proof.

Combine (3.13) and (3.31), we conclude that there is a unique solution
$(v_{1},v_{2},v_{4}(-1))\in C^{1,\alpha}\times\mathbb{R}$ such that

(3.34)
$\displaystyle\|(v_{1},v_{2})\|_{C^{\alpha}(\bar{Q})}+|{\textbf{v}}_{4}(-1)|\leq
C(\|(G_{1},G_{2})\|_{C^{\alpha}(\bar{Q})}$
$\displaystyle+\|(R_{6}(y_{2}),R_{7}(y_{2}))\|_{{}_{C^{1,\alpha}[-1,1]}}+\|\epsilon\hat{P}_{ex}(y_{2})\|_{C^{1,\alpha}[-1,1]}),$

The similar estimates also holds true for
$\|(v_{1},v_{2})\|_{C^{1,\alpha}(\bar{Q})}$, but we can derive an even better
estimate by rewriting (3.8) into an elliptic equation of $v_{1}$. To this end,
we firstly rewrite the system (3.8) as

(3.35)
$\displaystyle\begin{cases}\partial_{y_{1}}v_{1}+\frac{1}{1-\bar{M}^{2}}\partial_{y_{2}}v_{2}+\lambda_{1}(y_{1})v_{1}=\mathcal{G}_{1}(y),\\\
\partial_{y_{1}}v_{2}-\partial_{y_{2}}v_{1}=\mathcal{G}_{2}(y),\\\
v_{1}(L_{s},y_{2})=\mathcal{R}_{6}(y_{2}),\\\
v_{1}(L_{1},y_{2})=\mathcal{R}_{7}(y_{2}),\\\ v_{2}(y_{1},\pm
1)=0,\end{cases}$

where

$\displaystyle\mathcal{G}_{1}(y)=G_{1}(y)-\frac{B_{3}(y_{1})b_{3}+B_{4}(y_{1})}{(c^{2}(\bar{\rho}^{+})-(\bar{u}^{+})^{2})}(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau),$
$\displaystyle\mathcal{G}_{2}(y)=G_{2}(y)+b_{0}v_{2}(L_{s},y_{2}),$
$\displaystyle\mathcal{R}_{6}(y_{2})=b_{2}(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau)+R_{6}(y_{2}),$
$\displaystyle\mathcal{R}_{7}(y_{2})=\frac{b_{3}}{\bar{u}(L_{1})}(v_{4}(-1)+\int_{-1}^{y_{2}}b_{0}v_{2}(L_{s},\tau)d\tau)-\frac{\epsilon\hat{P}_{ex}(y_{2})}{\bar{u}(L_{1})}+R_{7}(y_{2}),$

and

(3.36) $\displaystyle\partial_{y_{2}}\mathcal{G}_{1}(y_{1},\pm
1)=0,\,\mathcal{G}_{2}(y_{1},\pm 1)=0,\,\mathcal{R}^{\prime}_{6}(\pm
1)=0,\,\mathcal{R}^{\prime}_{7}(\pm 1)=0.$

Note that (3.13) and the boundary conditions of $\phi$ implies that

$\displaystyle v_{2}(y_{1},\pm 1)=0,\,\partial_{y_{2}}v_{1}(y_{1},\pm 1)=0.$

A simple cancellation yields to

(3.37)
$\displaystyle\begin{cases}\partial_{y_{1}}((1-\bar{M}^{2})\partial_{y_{1}}v_{1})+\partial^{2}_{y_{2}}v_{1}+\partial_{y_{1}}(\lambda_{1}(1-\bar{M}^{2})v_{1})=\partial_{y_{1}}((1-\bar{M}^{2})\mathcal{G}_{1})-\partial_{y_{2}}\mathcal{G}_{2},\\\
v_{1}(L_{s},y_{2})=\mathcal{R}_{6}(y_{2}),\,v_{1}(L_{1},y_{2})=\mathcal{R}_{7}(y_{2}),\\\
\partial_{y_{2}}v_{1}(y_{1},\pm 1)=0,\end{cases}$

Since the boundary conditions in (3.37) are compatible at the corners, we
obtain the following estimates for $v_{1}$ by using the symmetric extension
defined by (3.25)

(3.38) $\displaystyle\|v_{1}\|_{C^{k,\alpha}(\bar{Q})}\leq$
$\displaystyle\,C(\|v_{2}\|_{C^{k-1,\alpha}(\bar{Q})}+|v_{4}(-1)|+\|(G_{1},G_{2})\|_{C^{k-1,\alpha}(\bar{Q})}$
$\displaystyle+\|\hat{P}_{ex}\|_{C^{k,\alpha}[-1,1]}+\|(R_{6},R_{7})\|_{C^{k,\alpha}[-1,1]}),\,k=1,2.$

This estimates together with (3.35) also implies that

(3.39) $\displaystyle\|v_{2}\|_{C^{k,\alpha}(\bar{Q})}\leq$
$\displaystyle\,C(\|v_{2}\|_{C^{k-1,\alpha}(\bar{Q})}+|v_{4}(-1)|+\|(G_{1},G_{2})\|_{C^{k-1,\alpha}(\bar{Q})}$
$\displaystyle+\|\hat{P}_{ex}\|_{C^{k,\alpha}[-1,1]}+\|(R_{6},R_{7})\|_{C^{k,\alpha}[-1,1]}),\,k=1,2.$

Hence, (3.32) is follows from (3.34) and (3.38)-(3.39). Finally,
differentiating the first equation in (3.35) with respect to $x_{2}$ and
combining with $\partial_{y_{2}}\mathcal{G}_{1}(y_{1},\pm 1)=0$ and
$\partial_{y_{2}}v_{1}(y_{1},\pm 1)=0$, we obtain

$\displaystyle\partial_{y_{2}}^{2}v_{2}(y_{1},\pm 1)=0,$

which gives the compatibility condition (3.33). Thus, the proof is complete. ∎

##  4\. A priori estimates and proofs of main results

In this section, we will use the Banach contraction mapping theorem to prove
Theorem 2.1. Given any $\hat{V}\in\mathcal{V}_{\delta}$, we could establish
some a priori estimates to the linearized problems defined in subsection 3.1,
and construct a contractible mapping from ${\mathcal{V}}_{\delta}$ into itself
so that there exits a unique fixed point, which is the solutions obtained in
Theorem 2.1 and the proof of Theorem 2.1 will be finished.

Lemma 3.4 implies that there is a unique solution $(v_{1},v_{2},v_{4}(-1))\in
C^{2,\alpha}(\bar{Q})\times\mathbb{R}$ to the system (3.8)-(3.9) satisfying

(4.1) $\displaystyle\|(v_{1},v_{2})\|_{C^{2,\alpha}(\bar{Q})}+|v_{4}(-1)|\leq
C(\|(G_{1},G_{2})\|_{C^{1,\alpha}(\bar{Q})}$
$\displaystyle+\|(R_{6}(y_{2}),R_{7}(y_{2}))\|_{{}_{C^{2,\alpha}[-1,1]}}+\|\epsilon\hat{P}_{ex}(y_{2})\|_{C^{2,\alpha}[-1,1]})$
$\displaystyle\leq C(\epsilon+\delta^{2}),$

and the compatibility condition (3.33) holds true.

The shock curve $v_{4}$ is given by (3.3), which satisfies

(4.2) $\displaystyle\|v_{4}\|_{C^{3,\alpha}[-1,1]}\leq
C(|v_{4}(-1)|+\|v_{2}\|_{C^{2,\alpha}(\bar{Q})}+\|F_{5}\|_{C^{2,\alpha}(\bar{Q})})\leq
C(\epsilon+\delta^{2}).$

Moreover, it follows from (3.33) and (3.4) that

(4.3) $\displaystyle v_{4}^{\prime}(\pm 1)=0=v_{4}^{(3)}(\pm 1).$

It remains to solve $v_{3}$. Due to (3.6), combining with (3.7) and (4.3), we
obtain the estimate

(4.4) $\displaystyle\|v_{3}\|_{C^{2,\alpha}(\bar{Q})}\leq
C(\|v_{4}\|_{C^{2,\alpha}[-1,1]}+\|F_{6}\|_{C^{2,\alpha}(\bar{Q})})\leq
C(\epsilon+\delta^{2}),$

and the compatibility condition

(4.5) $\displaystyle\partial_{y_{2}}v_{3}(y_{1},\pm 1)=0.$

Taking $\delta=O(1)\epsilon$, then for any given
$\hat{V}\in{\mathcal{V}}_{\delta}$ we can define a continuous mapping
$T:{\mathcal{V}}_{\delta}\rightarrow{\mathcal{V}}_{\delta}$ as

(4.6) $\displaystyle T\hat{V}=V,$

due to the iteration scheme introduced in the previous section and the
estimates (4.1)-(4.5). Finally, we show that the mapping is also contractible
in the space $(C^{1,\alpha}(\bar{Q}))^{3}\times C^{2,\alpha}[-1,1]$.

For arbitrarily given two states
$\hat{V}_{i}=(\hat{v}_{1}^{i},\hat{v}_{2}^{i},\hat{v}_{3}^{i},\hat{v}_{4}^{i})\in{\mathcal{V}}_{\delta},i=1,2$
with the corresponding fluid variable
$(\hat{u}_{1}^{i},\hat{u}_{2}^{i},\hat{B}^{i},\hat{\xi}^{i})$, set

$\displaystyle V_{i}=T\hat{V}_{i},\,i=1,2,$

where $V_{i}=(v_{1}^{i},v_{2}^{i},v_{3}^{i},v_{4}^{i})$. For the convenience,
we denote $\hat{W}=\hat{V}_{1}-\hat{V}_{2}$ and $W=V_{1}-V_{2}$, or
equivalently,

$\displaystyle\hat{w}_{k}=\hat{v}_{k}^{1}-\hat{v}_{k}^{2},\,\,w_{k}=v_{k}^{1}-v_{k}^{2},\,\,1\leq
k\leq 4.$

The equation (3.2) implies that

(4.7) $\displaystyle
w_{4}^{\prime}=b_{0}w_{2}+O(\epsilon)\sum_{i=1}^{4}\hat{w}_{i},$

which yields that

(4.8) $\displaystyle\|w_{4}^{\prime}\|_{C^{1,\alpha}[-1,1]}\leq
C\|w_{2}\|_{C^{1,\alpha}(\bar{Q})}+C\epsilon(\sum_{i=1}^{3}\|\hat{w}_{i}\|_{C^{1,\alpha}(\bar{Q})}+\|\hat{w}_{4}\|_{C^{1,\alpha}[-1,1]}).$

It follows from (3.6) that

(4.9) $\displaystyle
w_{3}=b_{3}w_{4}-b_{3}\int_{\beta_{1}(y)}^{\beta_{2}(y)}(\hat{v}_{4}^{1}(\tau))^{\prime}d\tau+b_{3}\int_{y_{2}}^{\beta_{2}(y)}\hat{w}_{4}^{\prime}(\tau)d\tau+O(\epsilon)\sum_{i=1}^{3}\hat{w}_{i},$

where $\beta_{i},i=1,2$ is the initial position such that the corresponding
characteristic $y_{2}^{i}(s,\beta_{i})$ going through $(y_{1},y_{2})$ with
$y_{2}^{i}(L_{s})=\beta_{i}$. It is easy to verify that

(4.10) $\displaystyle\|\beta_{1}(y)-\beta_{2}(y)\|_{C^{1,\alpha}(\bar{Q})}\leq
C(\|\hat{w}_{1}\|_{C^{1,\alpha}(\bar{Q})}+\|\hat{w}_{2}\|_{C^{1,\alpha}(\bar{Q})}+\|\hat{w}_{4}\|_{C^{2,\alpha}[-1,1]}),$

thus,

(4.11) $\displaystyle\|w_{3}\|_{C^{1,\alpha}(\bar{Q})}\leq
C\|w_{4}\|_{C^{1,\alpha}[-1,1]}+C\epsilon(\sum_{i=1}^{3}\|\hat{w}_{i}\|_{C^{1,\alpha}(\bar{Q})}+\|\hat{w}_{4}\|_{C^{2,\alpha}[-1,1]}).$

It is straightforward to show that $w_{1},w_{2}$ satisfies

(4.12)
$\displaystyle\begin{cases}\partial_{y_{1}}w_{1}+\frac{1}{1-\bar{M}^{2}}\partial_{y_{2}}w_{2}+\lambda_{1}(y_{1})w_{1}+\lambda_{2}(y_{1})(w_{4}(-1)+\int_{-1}^{y_{2}}b_{0}w_{2}(L_{s},\tau)d\tau)\\\
=\sum_{i=1}^{4}(O(\epsilon)\hat{w}_{i}+O(\epsilon)\nabla\hat{w}_{i})+O(\epsilon)(\beta_{1}-\beta_{2})+O(1)\int_{y_{2}}^{\beta_{2}(y)}\hat{w}_{4}^{\prime}(\tau)d\tau,\\\
\partial_{y_{1}}w_{2}-\partial_{y_{2}}[w_{1}-\lambda(y_{1})(w_{4}(-1)+\int_{-1}^{y_{2}}b_{0}w_{2}(L_{s},\tau)d\tau)]\\\
=\sum_{i=1}^{4}(O(\epsilon)\hat{w}_{i}+O(\epsilon)\nabla\hat{w}_{i})+O(\epsilon)\partial_{y_{2}}(\beta_{1}-\beta_{2})+O(1)\partial_{y_{2}}\int_{y_{2}}^{\beta_{2}(y)}\hat{w}_{4}^{\prime}(\tau)d\tau,\end{cases}$

and the boundary conditions

(4.13)
$\displaystyle\begin{cases}w_{1}(L_{s},y_{2})=b_{2}(\hat{w}_{4}(-1)+\int_{-1}^{y_{2}}b_{0}\hat{w}_{2}(L_{s},\tau)d\tau)+\sum_{i=1}^{4}O(\epsilon)\hat{w}_{i},\\\
w_{1}(L_{1},y_{2})=\frac{b_{3}}{\bar{u}(L_{1})}(\hat{w}_{4}(-1)+\int_{-1}^{y_{2}}b_{0}\hat{w}_{2}(L_{s},\tau)d\tau)+\sum_{i=1}^{4}O(\epsilon)\hat{w}_{i}\\\
+O(\epsilon)(\beta_{1}-\beta_{2})(L_{1},y_{2})+O(1)\int_{y_{2}}^{\beta_{2}(L_{1},y_{2})}\hat{w}_{4}^{\prime}(\tau)d\tau,\\\
w_{2}(y_{1},\pm 1)=0,\end{cases}$

where

$\displaystyle\lambda_{2}(y_{1})=\frac{B_{3}(y_{1})b_{3}+B_{4}(y_{1})}{c^{2}(\bar{\rho}^{+})-(\bar{u}^{+})^{2}}.$

Then, applying the estimate (3.32) to the system (4.12)-(4.13) with $k=1$ and
together with (4.10), we obtain

(4.14) $\displaystyle\|(w_{1},w_{2})\|_{C^{1,\alpha}(\bar{Q})}+|w_{4}(-1)|\leq
C\epsilon(\sum_{i=1}^{3}\|\hat{w}_{i}\|_{C^{1,\alpha}(\bar{Q})}+\|\hat{w}_{4}\|_{C^{2,\alpha}[-1,1]}).$

Finally, collecting all these estimates above leads to

(4.15)
$\displaystyle\sum_{i=1}^{3}\|w_{i}\|_{C^{1,\alpha}(\bar{Q})}+\|w_{4}\|_{C^{2,\alpha}[-1,1]}\leq
C\epsilon(\sum_{i=1}^{3}\|\hat{w}_{i}\|_{C^{1,\alpha}(\bar{Q})}+\|\hat{w}_{4}\|_{C^{2,\alpha}[-1,1]}).$

By (4.15), there is a small constant $\epsilon_{0}$ such that for all
$\epsilon\in(0,\epsilon_{0}]$, the mapping $T$ defined by (4.6) is
contractible in the Banach space $(C^{1,\alpha}(\bar{Q}))^{3}\times
C^{2,\alpha}[-1,1]$. Therefore, there exists a unique solution $V$ in
$(C^{1,\alpha}(\bar{Q}))^{3}\times C^{2,\alpha}[-1,1]$. Due to the Lemma 3.4
and the a priori estimates (4.2),(4.4), we know that $V$ also belongs to
${\mathcal{V}}_{\delta}$. It follows that $V$ satisfies the estimates (2.23).
Thus , the proof of Theorem 2.1 is complete. Theorem 1.2 is a direct inference
of Theorem 2.1. We omit the details.

Acknowledgement. Weng is partially supported by National Natural Science
Foundation of China 11701431, 11971307, 12071359.

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