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

Gui-Qiang G. Chen  Gui-Qiang G. Chen: Mathematical Institute, University of
Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK;
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
<EMAIL_ADDRESS>,  Yun Pu  Yun Pu: Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100190, China; School of
Mathematical Sciences, Fudan University, Shanghai 200433, China
<EMAIL_ADDRESS>and  Yongqian Zhang  Yongqian Zhang: School of Mathematical
Sciences, Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>
###### Abstract.

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

###### Key words and phrases:

Inverse problem, supersonic flow, potential flow, Mach number, Euler
equations, perturbed cones, cone surface, Lipschitz cones, free boundary,
pressure distribution, BV, Glimm schemes, Glimm-type functional, self-similar
solutions, interaction estimates, reflection coefficients, asymptotic
behavior.

###### 2020 Mathematics Subject Classification:

35B07, 35B20, 35D30, 35L65, 35L67; 76J20, 76L05, 76N10

###### Contents

  1. 1 Introduction
  2. 2 Riemann Problems and Self-Similar Solutions of the Unperturbed Conic Flow
  3. 3 Approximate Solutions
  4. 4 Riemann-Type Problems and Interaction Estimates
  5. 5 Glimm-Type Functional and Compactness of the Approximate Solutions
  6. 6 Convergence of the Approximate Solutions
  7. 7 Asymptotic Behavior of Global Entropy Solutions

##  1\. Introduction

Fig. 1.1. The strong straight-sided conical shock

We are interested in the structural stability of inverse problems for the
three-dimensional (3-D) steady supersonic potential flows past a Lipschitz
perturbed cone with given states of the incoming flow together with Lipschitz
perturbed pressure distributions on its surface. The shock stability problem
of steady supersonic flows past Lipschitz cones is fundamental for the
mathematical theory of the multidimensional (M-D) hyperbolic systems of
conservation laws, since its solutions are time-asymptotic states and global
attractors of general entropy solutions of time-dependent initial-boundary
value problems (IBVP) with abundant nonlinear phenomena, besides its
significance to many fields of applications including aerodynamics; see [3, 9,
29, 20] and references cited therein. Meanwhile, the corresponding inverse
problems play essential roles in airfoil design; see [1, 2, 5, 26, 27, 28, 39,
40, 43, 45]. As indicated in [20], when a uniform supersonic flow of constant
speed from the far-field (negative infinity) hits a straight cone, given a
constant pressure distribution that is less than a critical value on the cone
surface, the vertex angle of the cone can be determined such that there is a
supersonic straight-sided conical shock attached to the cone vertex, and the
state between the conical shock-front and the cone can be obtained by the
shooting method, which is a self-similar solution; see Fig. 1.1. In this
paper, we focus our analysis on the stability of an inverse problem, along
with the background self-similar solutions, in the steady potential flows that
are axisymmetric with respect to the $x$–axis, given the pressure
distributions of gas on the cones, whose boundary surfaces in
$\mathbb{R}^{3}$, formed by the rotation of generating curves of form
$\Gamma:=\\{(x,b(x))\,:\,x\geq 0\\}$ around the $x$–axis, are to be
determined; see Fig. 1.2.

Fig. 1.2. Supersonic flow past an axisymmetric cone

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

$\left\\{\begin{aligned} &(\rho u)_{x}+(\rho v)_{y}=-\frac{\rho v}{y},\\\
&v_{x}-u_{y}=0,\end{aligned}\right.$ (1.1)

together with the Bernoulli law:

$\frac{u^{2}+v^{2}}{2}+\frac{c^{2}}{\gamma-1}=\frac{u_{\infty}^{2}}{2}+\frac{c_{\infty}^{2}}{\gamma-1},$
(1.2)

where $U:=(u,v)^{\top}$ is the velocity in the $(x,y)$–coordinates, $\rho$ is
the flow density, and $U_{\infty}=(u_{\infty},0)^{\top}$ and $\rho_{\infty}$
are the velocity and the density of the incoming flow, respectively. The
Bernoulli law in (1.2) is obtained via the constitutive relation between
pressure $p$ and density $\rho$: $p=\rho^{\gamma}$, with $\gamma>1$ for the
polytropic isentropic gas and $\gamma=1$ for the isothermal flow, under
scaling. In particular, $c=:\sqrt{\frac{\gamma p}{\rho}}$ is called the sonic
speed, and $M:=\sqrt{\frac{u^{2}+v^{2}}{c^{2}}}$ is called the Mach number.

The Bernoulli law (1.2) can be written as

$\frac{u^{2}+v^{2}}{2}+\frac{(u^{2}+v^{2})M^{-2}}{\gamma-1}=\frac{u_{\infty}^{2}}{2}+\frac{M_{\infty}^{-2}}{\gamma-1}.$
(1.3)

Without loss of generality, we may choose $u_{\infty}=1$ by scaling;
otherwise, we can simply scale: $U\rightarrow u_{\infty}^{-1}U$, in system
(1.1) and (1.3). With the fixed $u_{\infty}=1$, $M_{\infty}\to\infty$ is
equivalent to $p_{\infty}\to 0$, or $c_{\infty}\to 0$.

System (1.1) can be written in the form:

$\partial_{x}W(U)+\partial_{y}H(U)=E(U,y)$ (1.4)

with $U=(u,v)^{\top}$, where

$\displaystyle W(U)=(\rho u,v)^{\top},\quad\,\,H(U)=(\rho
v,-u)^{\top},\quad\,\,E(U,y)=(-\frac{\rho v}{y},0)^{\top},$

and $\rho$ is a function of $U$ determined by the Bernoulli law (1.2).

When $\rho>0$ and $u>c$, $U$ can also be presented by $W(U)=(\rho
u,v)^{\top}$, i.e., $U=U(W)$, by the implicit function theorem, since the
Jacobian:

$\displaystyle\det\big{(}\nabla_{U}W(U)\big{)}=-\frac{\rho}{c^{2}}(u^{2}-c^{2})<0.$

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

$\partial_{x}W+\partial_{y}H(U(W))=E(U(W),y).$ (1.5)

Therefore, system (1.1)–(1.2) becomes a hyperbolic system of conservation laws
with source terms of form (1.5). Such nonhomogeneous hyperbolic systems of
conservation laws also arise naturally in other problems from many important
applications, which exhibit rich phenomena; for example, see [9, 11, 12, 13,
20, 23] and the references cited therein.

Throughout this paper, the following conditions are assumed:

(A1) $\,p^{b}(x)>0$ for $x>0$,

$\displaystyle p^{b}(x)=p_{0}\qquad\mbox{for $x\in[0,x_{0}]$},$

where $x_{0}>0$, $p_{0}\in(0,p^{*})$ for some $p^{*}>0$ to be determined by
$\gamma>1$, and

$\displaystyle(p^{b})^{\prime}_{+}(x)=\lim_{t\rightarrow
x+}\frac{p^{b}(t)-p^{b}(x)}{t-x}\in\text{BV}([0,\infty)).$

(A2) The velocity of the incoming flow $U_{\infty}=(1,0)^{\top}$ is
supersonic: $M_{\infty}>1$.

Given a perturbed pressure distribution $p^{b}(x)$ on the cone surface, the
problem is axisymmetric with respect to the $x-$axis. Thus, it suffices to
analyze the problem in the half-space $\\{y\leq 0\\}$. Then the inverse
problem is to find the generating curve $y=b(x)\leq 0$ of the cone surface and
a global solution in the domain:

$\Omega=\big{\\{}(x,y)\,:\,x\geq 0,y<b(x)\big{\\}}$ (1.6)

with its upper boundary:

$\Gamma=\big{\\{}(x,y)\,:\,x\geq 0,y=b(x)\big{\\}}$ (1.7)

such that

$U\cdot\textbf{n}|_{\Gamma}=0,$ (1.8)

where
$\textbf{n}=\textbf{n}\big{(}x,b(x)\big{)}=\frac{(-b^{\prime}(x),1)^{\top}}{\sqrt{1+(b^{\prime}(x))^{2}}}$
is the corresponding outer normal vector to $\Gamma$ at a differentiable point
$(x,b(x))\in\Gamma$.

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

Cauchy Condition:

$U|_{x=0}=U_{\infty}:=(1,0)^{\top},$ (1.9)

Boundary Condition:

$p\big{(}x,b(x)\big{)}=p^{b}(x).$ (1.10)

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

Definition 1.1. (Entropy Solutions). Consider the inverse problem
(1.5)–(1.10). A function $b(x)\in\text{Lip}([0,\infty))$ is called a
generating curve $\Gamma$ of the cone surface as defined in (1.7), and a
vector function
$U=(u,v)^{\top}\in(\text{BV}_{\text{loc}}\cap\text{L}^{\infty})(\Omega)$ with
$\Omega$ defined in (1.6) is called an entropy solution of (1.5)–(1.10) if
they satisfy the following conditions:

  * (i)

For any test function $\phi\in\text{C}_{0}^{1}(\mathbb{R}^{2};\mathbb{R})$ and
$\psi\in\text{C}_{0}^{1}(\Omega;\mathbb{R})$,

$\displaystyle\iint_{\Omega}\big{(}\rho u\phi_{x}+\rho v\phi_{y}-\dfrac{\rho
v}{y}\phi\big{)}\,{\rm d}x{\rm
d}y+\int_{-\infty}^{0}\phi(0,y)\rho_{\infty}u_{\infty}\,{\rm d}y=0,$ (1.11)
$\displaystyle\iint_{\Omega}\left(v\psi_{x}-u\psi_{y}\right){\rm d}x{\rm
d}y=0,$

  * (ii)

For any convex entropy pair $(\mathcal{E},\mathcal{Q})$ with respect to $W$ of
(1.5), i.e., $\nabla^{2}\mathcal{E}(W)\geq 0$ and
$\nabla\mathcal{Q}(W)=\nabla\mathcal{E}(W)\nabla H(U(W))$,

$\displaystyle\iint_{\Omega}\big{(}\mathcal{E}(W(U))\varphi_{x}+\mathcal{Q}(W(U))\varphi_{y}+\nabla_{W}\mathcal{E}(W(U))E(U,y)\varphi\big{)}\,{\rm
d}x{\rm d}y$ (1.12)
$\displaystyle+\int_{-\infty}^{0}\mathcal{E}(W(U_{\infty}))\varphi(0,y)\,{\rm
d}y\geq 0$

for any $\varphi\in\text{C}_{0}^{1}(\Omega;\mathbb{R})$ with $\varphi\geq 0$.

Remark 1.1. For the potential flow, the Bernoulli law (1.2) gives

$\displaystyle\frac{M^{2}}{2}+\frac{1}{\gamma-1}=\frac{B_{\infty}}{c^{2}}$

for $B_{\infty}=\frac{u_{\infty}^{2}}{2}+\frac{c_{\infty}^{2}}{\gamma-1}$,
$c^{2}=\gamma\rho^{\gamma-1}$, and $p=\rho^{\gamma}$. Then the assumptions on
pressure $p^{b}$ can be reduced to the equivalent ones on the Mach number
$M_{b}$ on the unknown boundary $\Gamma$.

Main Theorem (Existence and stability). Let (1.2)–(1.2) hold, and let
$1<\gamma<3$ and

$0<p_{0}<p^{*}:=\Big{(}(\sqrt{\gamma+7}-\sqrt{\gamma-1})\sqrt{\frac{\gamma-1}{16\gamma}}\Big{)}^{\frac{2\gamma}{\gamma-1}}.$

Assume that $M_{\infty}$ is sufficiently large and $\epsilon_{0}$ is
sufficiently small such that

$\displaystyle\text{T.V.}\,p^{b}<\epsilon_{0},$ (1.13)

then the following statements hold:

  * (i)

Global existence: IBVP (1.4)–(1.10) determines a boundary
$y=b(x)=\int_{0}^{x}b^{\prime}(t)\,{\rm d}t$ with
$b^{\prime}(x)\in$BV$(\mathbb{R}_{+})$, which is a small perturbation of the
generating curve of the straight-sided cone surface: $y=b_{0}x$, and admits a
global entropy solution $U(x,y)$ with bounded total variation:

$\sup_{x>0}\text{T.V.}\\{U(x,y)\,:\,-\infty<y<b(x)\\}<\infty$ (1.14)

in the sense of Definition 1.2, containing a strong leading shock-front
$y=\chi(x)=\int_{0}^{x}s(t){\rm d}t$ with $s(x)\in$BV$(\mathbb{R}_{+})$ which
is a small perturbation of the strong straight-sided conical shock-front
$y=s_{0}x$, so that the solution between the leading shock-front and the cone
surface is a small perturbation of the background self-similar solution of the
straight-sided cone case, where $s_{0}$ denotes the slope of the corresponding
straight-sided shock-front, and $b_{0}$ is the slope of the generating curve
of the straight-sided cone surface.

  * (ii)

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

$\lim_{x\rightarrow\infty}\sup\left\\{\big{|}U_{\vartheta}(x,y)-\tilde{U}(\sigma;s_{\infty},G(s_{\infty}))\big{|}\,:\,\,\chi_{\vartheta}(x)<y<b_{\vartheta}(x)\right\\}=0$
(1.15)

with $\tilde{U}(\sigma;s_{\infty},G(s_{\infty}))$ satisfying
$\tilde{U}(s_{\infty};s_{\infty},G(s_{\infty}))=G(s_{\infty})$,

$\displaystyle\tilde{U}(b_{\infty}^{\prime};s_{\infty},G(s_{\infty}))\cdot(-b_{\infty}^{\prime},1)=0,$
(1.16)
$\displaystyle\dfrac{1}{2}\big{|}\tilde{U}(b_{\infty}^{\prime};s_{\infty},G(s_{\infty}))\big{|}^{2}+\dfrac{\gamma(p^{b}_{\infty})^{\frac{\gamma-1}{\gamma}}}{\gamma-1}=\dfrac{1}{2}+\dfrac{\gamma
p_{\infty}^{\frac{\gamma-1}{\gamma}}}{\gamma-1},$

where

$\displaystyle
p^{b}_{\infty}=\lim_{x\rightarrow\infty}p^{b}(x),\quad\,\,s_{\infty}=\lim_{x\rightarrow\infty}s_{\vartheta}(x),\quad\,\,b^{\prime}_{\infty}=\lim_{x\rightarrow\infty}(b_{\vartheta})^{\prime}_{+}(x),$
(1.17)

$\tilde{U}(\sigma;s,G(s))$ is the state of the self-similar solution, and
$G(s)$ denotes the state connected to state $U_{\infty}$ by the strong leading
shock-front of speed $s$.

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

When the surface of the perturbed cone is smooth, using the weighted energy
methods, Chen-Xin-Yin established the global existence of piecewise smooth
solutions in [19]. They considered a 3-D axisymmetric potential flow past a
symmetrically perturbed cone under the assumption that the attached angle is
sufficiently small and the Mach number of the incoming flow is sufficiently
large. This result was also extended to the M-D potential flow case; see [32]
for more details. Under a certain boundary condition on the cone surface, the
global existence of the M-D conical shock solutions was obtained in [49] when
the uniform supersonic incoming flow with large Mach number past a generally
curved sharp cone. Meanwhile, using a delicate expansion of the background
solution, Cui-Yin established the global existence and stability of a steady
conical shock wave in [21, 22] for the symmetrically perturbed supersonic flow
past an infinitely long cone whose vertex angle is less than the critical
angle. More recently, by constructing new background solutions that allow the
speeds of the incoming flows to approach the limit speed, the global existence
of steady symmetrically conical shock solutions was established in Hu-Zhang
[29] when a supersonic incoming potential flow hits a symmetrically perturbed
cone with an opening angle less than the critical angle. We also remark that
some pivotal results have been obtained on the stability of M-D transonic
shocks under symmetric perturbations of the straight-sided cones or the
straight-sided wedges, as well as on Radon measure solutions for steady
compressible Euler equations of hypersonic-limit conical flows; see [6, 7, 8,
42, 50] and the references cited therein.

Corresponding to these shock stability problems, two types of inverse problems
have been considered. One type is for the problems of determining the shape of
the wedge in the planar steady supersonic flow for the given location of the
leading shock front. This kind of inverse problems and the related inverse
piston problems have been considered by Li-Wang in [35, 37, 34, 36, 46, 47],
where the leading shock-front is assumed to be smooth and the characteristic
method is applied to find the piecewise smooth solution with the leading shock
as its only discontinuity; see also [33]. The other one is for the problems of
determining the shape of the wedge or the cone with given pressure
distribution on it in the planar steady supersonic flow (cf. [41]) or
axisymmetric conical steady supersonic flow. Though various numerical methods
and the linearized method have been proposed to deal with this type of
problems, there seems no rigorous result on the existence of solutions to such
inverse problems for steady supersonic flow past a cone.

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

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

To overcome the new difficulties, we first fix the center of the self-similar
variable to be the origin when solving the differential equations (2.3) and
then develop a modified Glimm scheme to construct approximate solutions
$U_{\Delta x,\vartheta}(x,y)$ via the self-similar solutions as building
blocks in order to incorporate the geometric source term. In our construction,
the grid points are fixed at the beginning, which are the intersections of
lines $x=x_{h}$, $h\in\mathbb{N}$, and the rays issuing from the origin (the
vertex point of the cone). Consequently, this construction allows us to find
new terms $\theta_{b}(h)$ to control the increasing part of the Glimm type
functional near the approximate boundary (see Lemma 4.2), while it brings us
an extra error so that the boundary conditions on the approximate boundary are
no longer satisfied everywhere, but are satisfied at the initial point of each
approximate boundary at each step. Nevertheless, in Proposition 6, we are able
to prove that this error goes to zero as the grid size $\Delta x$ tends to
zero.

Furthermore, we make careful asymptotic expansions of the self-similar
solutions with respect to $M_{\infty}^{-1}$. We then make full use of the
asymptotic expansion analysis of the background solutions with respect to
$M_{\infty}^{-1}$ to calculate the reflection coefficients $K_{r,1}$,
$K_{w,2}$, $K_{s}$, and $\mu_{w,2}$ of the weak waves reflected from both the
boundary and the strong leading shock, and the self-similar solutions
reflected from the strong leading shock to prove that

$\lim_{M_{\infty}\rightarrow\infty}\big{(}|K_{r,1}||K_{w,2}|+|K_{r,1}||K_{s}||\mu_{w,2}|\big{)}<1.$

Based on this, we choose some appropriate weights, independent of
$M_{\infty}$, in the construction of the Glimm-type functional and show that
the functional is monotonically decreasing. Then the convergence of the
approximate solutions is followed by the standard approach for the Glimm-type
scheme as in [24, 31]; see also [14, 18, 4, 23, 44]. Finally, the existence of
entropy solutions and the asymptotic behavior of the entropy solutions are
also proved.

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

##  2\. Riemann Problems and Self-Similar Solutions of the Unperturbed Conic
Flow

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

$\left\\{\begin{aligned} &(\rho u)_{x}+(\rho v)_{y}=0,\\\
&v_{x}-u_{y}=0,\end{aligned}\right.$ (2.1)

is strictly hyperbolic with two distinctive eigenvalues:

$\displaystyle\lambda_{i}=\frac{uv+(-1)^{i}c\sqrt{u^{2}+v^{2}-c^{2}}}{u^{2}-c^{2}},\qquad
i=1,\ 2,$

for $u>c_{*}$ and $u^{2}+v^{2}<q_{*}^{2}$, where

$\displaystyle
c_{*}=\sqrt{\frac{\gamma-1}{\gamma+1}+\frac{2c_{\infty}^{2}}{\gamma+1}},\quad\,q_{*}=\sqrt{1+\frac{2c_{\infty}^{2}}{\gamma+1}}.$

Denote $q:=\sqrt{u^{2}+v^{2}}$ and $\theta:=\arctan\frac{v}{u}$. Then

$\displaystyle\lambda_{i}=\tan(\theta+(-1)^{i}\theta_{m}),\qquad i=1,2,$

where

$\displaystyle\theta_{m}:=\arctan\frac{c}{\sqrt{q^{2}-c^{2}}}$

is the Mach angel. A direct computation indicates
$\theta_{m}\in(0,\frac{\pi}{2})$.

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

Lemma 2.1. For $u>c_{*}$ and $q<q_{*}$,

$\displaystyle(-\lambda_{i},1)\cdot(\frac{\partial\lambda_{i}}{\partial
u},\frac{\partial\lambda_{i}}{\partial
v})=\frac{\gamma+1}{2\sqrt{q^{2}-c^{2}}}\sec^{3}(\theta+(-1)^{i}\theta_{m}),\quad\,i=1,2.$

Then, setting

$\displaystyle
r_{i}(U)=\frac{(-\lambda_{i}(U),1)}{(-\lambda_{i}(U),1)\cdot\nabla\lambda_{i}(U)},\quad\,i=1,2,$

we see that $r_{i}(U)\cdot\nabla\lambda_{i}(U)=1$ for $i=1,2$.

Denote the supersonic part of the shock polar by

$S((u_{\infty},0))=\big{\\{}(\bar{u},\bar{v})\,:\,\bar{c}^{2}<\bar{u}^{2}+\bar{v}^{2}\leq
1\big{\\}},$

where $(\bar{u},\bar{v})$ satisfies the Rankine–Hugoniot condition:

$\left\\{\begin{aligned} &\bar{\rho}(\bar{u}s-\bar{v})=\rho_{\infty}s,\\\
&\bar{u}+\bar{v}s=1,\end{aligned}\right.$ (2.2)

with
$\frac{\bar{u}^{2}+\bar{v}^{2}}{2}+\frac{\gamma\bar{\rho}^{\gamma-1}}{\gamma-1}=\frac{1}{2}+\frac{c_{\infty}^{2}}{\gamma-1}$.
Let

$S_{1}^{-}((u_{\infty},0))=\big{\\{}(\bar{u},\bar{v})\,:\,(\bar{u},\bar{v})\in
S((u_{\infty},0)),\,\bar{v}<0\big{\\}}$

be the part of shock polar corresponding to the $\lambda_{1}$–characteristic
field. Similar to [30, 51, 52], we can parameterize the shock polar
$S_{1}^{-}((u_{\infty},0))$ by a C2–function:

$G:s\mapsto G(s;U_{\infty})\qquad\,\,\mbox{with $U_{\infty}=(1,0)$},$

where $G(s;U_{\infty})$ is a supersonic state connected with $U_{\infty}$ by a
shock of speed $s$. For simplicity, we write $G(s;U_{\infty})$ as $G(s)$ and
use $\bar{u}(s)$ and $\bar{v}(s)$ to denote the components of $G(s)$, that is,
$G(s)=(\bar{u}(s),\bar{v}(s))^{\top}$. Then we have the following property for
$S_{1}^{-}((u_{\infty},0))$ (cf. [16]).

Lemma 2.2. For $s<\lambda_{1}(U_{\infty})$, $\,\bar{\rho}(s)$ is a strictly
monotonically decreasing function of $s$, and $\bar{u}(s)$ is a strictly
monotonically increasing function of $s$.

As in [20], let $\sigma=\frac{y}{x}$. Then the equations in (1.1) become

$\left\\{\begin{aligned}
&\big{(}1-\frac{u^{2}}{c^{2}}\big{)}\sigma^{2}u_{\sigma}-\frac{2uv}{c^{2}}\sigma^{2}v_{\sigma}-\big{(}1-\frac{v^{2}}{c^{2}}\big{)}\sigma
v_{\sigma}-v=0,\\\ &u_{\sigma}+\sigma v_{\sigma}=0.\end{aligned}\right.$ (2.3)

or, equivalently,

$\left\\{\begin{aligned}
&u_{\sigma}=\frac{c^{2}v}{(1+\sigma^{2})c^{2}-(v-\sigma u)^{2}},\\\
&v_{\sigma}=-\frac{c^{2}v}{\sigma\big{(}(1+\sigma^{2})c^{2}-(v-\sigma
u)^{2}\big{)}},\\\ &\rho_{\sigma}=\frac{\rho v(v-\sigma
u)}{\sigma\big{(}(1+\sigma^{2})c^{2}-(v-\sigma
u)^{2}\big{)}}.\end{aligned}\right.$ (2.4)

Given a constant state $(\bar{u},\bar{v})=G(s)$ on
$S_{1}^{-}((u_{\infty},0))$, there exists a local solution
$\tilde{U}(\sigma;s,G(s))=(\tilde{u}(\sigma;s),\tilde{v}(\sigma;s))$ of system
(2.3) with initial data:

$\displaystyle(\tilde{u}(s;s),\,\tilde{v}(s;s))=(\bar{u},\,\bar{v}).$

This solution can be extended to an end-point
$(\tilde{u}(\sigma_{e};s),\,\tilde{v}(\sigma_{e};s))$ with
$\frac{\tilde{v}(\sigma_{e};s)}{\tilde{u}(\sigma_{e};s)}=\sigma_{e}$. As
$(\bar{u},\bar{v})$ varies on $S_{1}^{-}((u_{\infty},0))$, the collection of
these end-states forms an apple curve (Fig. 2.1) through $U_{\infty}$; see
[20]. For these solutions, we have following properties, whose proof can be
found in [48].

Lemma 2.3. For $\tilde{u}(s;s)>\tilde{c}(s;s)$ and $\sigma\in(s,\sigma_{e})$,
then $\tilde{u}(\sigma;s)\sigma-\tilde{v}(\sigma;s)<0$,

$\displaystyle\frac{\partial\tilde{u}}{\partial\sigma}<0,\qquad\frac{\partial\tilde{v}}{\partial\sigma}<0,$
$\displaystyle\tilde{c}(\sigma;s)-\frac{\tilde{v}(\sigma;s)-\sigma\tilde{u}(\sigma;s)}{\sqrt{1+\sigma^{2}}}>\tilde{c}(s;s)-\frac{\tilde{v}(s;s)-s\tilde{u}(s;s)}{\sqrt{1+s^{2}}}>0,$

with
$\frac{\tilde{u}^{2}+\tilde{v}^{2}}{2}+\frac{\tilde{c}^{2}}{\gamma-1}=\frac{1}{2}+\frac{c_{\infty}^{2}}{\gamma-1}$.

Thus, we obtain the following estimate of the self-similar solution
$(\tilde{u}(\sigma;s),\,\tilde{v}(\sigma;s))$.

Lemma 2.4. For $\tilde{u}(s;s)>\tilde{c}(s;s)$ and $\sigma\in(s,\sigma_{e})$,

$\displaystyle\frac{1}{1+s^{2}}<\tilde{u}(\sigma;s)<\tilde{u}(s;s),\qquad\tilde{c}(s;s)<\tilde{c}(\sigma;s)<\tilde{c}(\sigma_{e};s)<\sqrt{\frac{(\gamma-1)s^{2}}{2(1+s^{2})}+\frac{1}{M_{\infty}^{2}}}.$

Fig. 2.1. Apple curve

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

$p^{*}=\left(\big{(}\sqrt{\gamma+7}-\sqrt{\gamma-1}\big{)}\sqrt{\frac{\gamma-1}{16\gamma}}\right)^{\frac{2\gamma}{\gamma-1}}.$
(2.5)

Lemma 2.5. Let $1<\gamma<3$ and $p_{0}\in(0,p^{*})$. For $M_{\infty}$ large
enough, the equations:

$\left\\{\begin{aligned}
&\rho_{0}(u^{\sharp}s^{\sharp}-v^{\sharp})=\rho_{\infty}s^{\sharp},\\\
&u^{\sharp}+v^{\sharp}s^{\sharp}=1,\\\
&\frac{(u^{\sharp})^{2}+(v^{\sharp})^{2}}{2}+\frac{c_{0}^{2}}{\gamma-1}=\frac{1}{2}+\frac{c_{\infty}^{2}}{\gamma-1}\end{aligned}\right.$
(2.6)

have a unique solution $(u^{\sharp},v^{\sharp},s^{\sharp})$ with
$s^{\sharp}<0$, where $\rho_{\infty}=p_{\infty}^{\frac{1}{\gamma}}$,
$\rho_{0}=p_{0}^{\frac{1}{\gamma}}$, and
$c_{0}=\sqrt{\gamma}p_{0}^{\frac{\gamma-1}{2\gamma}}$, such that

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}u^{\sharp}$
$\displaystyle=\lim_{M_{\infty}\rightarrow\infty}u_{a}=1-\frac{2c_{0}^{2}}{\gamma-1},$
(2.7) $\displaystyle\lim_{M_{\infty}\rightarrow\infty}v^{\sharp}$
$\displaystyle=\lim_{M_{\infty}\rightarrow\infty}v_{a}=-\sqrt{\frac{2c_{0}^{2}}{\gamma-1-2c_{0}^{2}}}\Big{(}1-\frac{2c_{0}^{2}}{\gamma-1}\Big{)},$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}c_{a}^{2}$
$\displaystyle=c_{0}^{2}.$

where

$u_{a}:=\frac{1}{1+(s^{\sharp})^{2}},\quad
v_{a}:=\frac{s^{\sharp}}{1+(s^{\sharp})^{2}},\quad
c_{a}:=\sqrt{\frac{(\gamma-1)(s^{\sharp})^{2}}{2(1+(s^{\sharp})^{2})}+\frac{1}{M_{\infty}^{2}}},$
(2.8)

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

$u^{\sharp}=\frac{\rho_{0}+\rho_{\infty}(s^{\sharp})^{2}}{\rho_{0}(1+s^{2})},\qquad
v^{\sharp}=\frac{(\rho_{0}-\rho_{\infty})s^{\sharp}}{\rho_{0}\big{(}1+(s^{\sharp})^{2}\big{)}}.$
(2.9)

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

$\displaystyle\bigg{(}\frac{\rho_{0}+\rho_{\infty}(s^{\sharp})^{2}}{\rho_{0}\big{(}1+(s^{\sharp})^{2}\big{)}}\bigg{)}^{2}+\bigg{(}\frac{(\rho_{0}-\rho_{\infty})s^{\sharp}}{\rho_{0}\big{(}1+(s^{\sharp})^{2}\big{)}}\bigg{)}^{2}=1+\frac{2(c_{\infty}^{2}-c_{0}^{2})}{\gamma-1},$

which gives

$\displaystyle(s^{\sharp})^{2}$
$\displaystyle=\frac{2(c_{0}^{2}-M_{\infty}^{-2})\rho_{0}^{2}}{(\gamma-1)\big{(}\rho_{0}^{2}-(\gamma
M_{\infty}^{2})^{-\frac{2}{\gamma-1}}\big{)}-2(c_{0}^{2}-M_{\infty}^{-2})\rho_{0}^{2}}$
$\displaystyle=\frac{2c_{0}^{2}}{\gamma-1-2c_{0}^{2}}-\frac{2(\gamma-1)}{(\gamma-1-2c_{0}^{2})^{2}}\,M_{\infty}^{-2}+O(M_{\infty}^{-\frac{4}{\gamma-1}})\qquad\text{as
$M_{\infty}\rightarrow\infty$}.$

Therefore, noting that $s^{\sharp}<0$, we obtain

$\displaystyle
s^{\sharp}=-\sqrt{\frac{2c_{0}^{2}}{\gamma-1-2c_{0}^{2}}}\bigg{(}1-\frac{\gamma-1}{2c_{0}^{2}\left(\gamma-1-2c_{0}^{2}\right)}\,M_{\infty}^{-2}\bigg{)}+O(M_{\infty}^{-\frac{4}{\gamma-1}})\qquad\text{as
$M_{\infty}\rightarrow\infty$}.$

Substituting the above expansion into (2.8)–(2.9), yields (2.7). $\square$

Lemma 2.6. Let $1<\gamma<3$ and $p_{0}\in(0,p^{*})$. For $M_{\infty}$ large
enough, there exists $\rho_{d}$ such that

$\rho_{0}^{\gamma-1}=\frac{\rho_{d}^{\gamma+1}-\rho_{\infty}^{\gamma+1}}{\rho_{d}^{2}-\rho_{\infty}^{2}},$
(2.10)

and the equations:

$\left\\{\begin{aligned} &\rho_{d}(u_{d}s_{d}-v_{d})=\rho_{\infty}s_{d},\\\
&u_{d}+v_{d}s_{d}=1,\\\
&\frac{u_{d}^{2}+v_{d}^{2}}{2}+\frac{c_{d}^{2}}{\gamma-1}=\frac{1}{2}+\frac{c_{\infty}^{2}}{\gamma-1},\end{aligned}\right.$
(2.11)

have a unique solution $(u_{d},v_{d},s_{d})$ with $s_{d}<0$ and
$c_{d}=\sqrt{\gamma\rho_{d}^{\gamma-1}}$. Moreover, for

$u^{\flat}:=\frac{1}{1+s_{d}^{2}},\quad\,v^{\flat}:=\frac{s_{d}}{1+s_{d}^{2}},$
(2.12)

we have

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}c_{d}^{2}$
$\displaystyle=c_{0}^{2},$ (2.13)
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}u^{\flat}$
$\displaystyle=\lim_{M_{\infty}\rightarrow\infty}u_{d}=1-\frac{2c_{0}^{2}}{\gamma-1},$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}v^{\flat}$
$\displaystyle=\lim_{M_{\infty}\rightarrow\infty}v_{d}=-\sqrt{\frac{2c_{0}^{2}}{\gamma-1-2c_{0}^{2}}}\,\Big{(}1-\frac{2c_{0}^{2}}{\gamma-1}\Big{)}.$

Proof. For each $\rho_{0}$, it is direct to find $\rho_{d}$ such that (2.10)
holds. Moreover, $p_{d}\in(0,p^{*})$ when $M_{\infty}$ is large enough. By
Lemma 2.1, we obtain the unique solution of (2.11):

$s_{d}=\frac{2(c_{0}^{2}-c_{\infty}^{2})}{\gamma-1-2(c_{0}^{2}-c_{\infty}^{2})},\quad\,u_{d}=\frac{\rho_{d}+\rho_{\infty}s_{d}^{2}}{\rho_{d}(1+s_{d}^{2})},\quad\,v_{d}=\frac{(\rho_{d}-\rho_{\infty})s_{d}}{\rho_{d}(1+s_{d}^{2})}.$
(2.14)

Then we have

$\displaystyle
s_{d}=-\sqrt{\frac{2c_{0}^{2}}{\gamma-1-2c_{0}^{2}}}\Big{(}1-\frac{\gamma-1}{2c_{0}^{2}(\gamma-1-2c_{0}^{2})}\,M_{\infty}^{-2}\Big{)}+O(M_{\infty}^{-4})\qquad\text{as
$M_{\infty}\rightarrow\infty$}.$

Substituting the above expansion into (2.12)–(2.14) yields (2.13). $\square$

Given $p_{0}\in(0,p^{*})$, we now solve the following problem for
$s_{0}<\sigma<b_{0}$:

$\begin{cases}\sigma^{2}\big{(}1-\frac{\tilde{u}^{2}}{\tilde{c}^{2}}\big{)}\tilde{u}_{\sigma}-\frac{2\tilde{u}\tilde{v}\sigma^{2}}{c^{2}}\tilde{v}_{\sigma}-\big{(}1-\frac{\tilde{v}^{2}}{\tilde{c}^{2}}\big{)}\tilde{v}_{\sigma}\sigma-\tilde{v}=0,\\\
\tilde{u}_{\sigma}+\sigma\tilde{v}_{\sigma}=0,\end{cases}$ (2.15)

with the boundary conditions:

$\displaystyle\,\,\,\tilde{\rho}(\tilde{u}s_{0}-\tilde{v})=\rho_{\infty}s_{0},\quad\tilde{u}+\tilde{v}s_{0}=1\qquad\,\,$
$\displaystyle\mbox{for $\sigma=s_{0}$},$ (2.16)
$\displaystyle\,\,\,\tilde{\rho}=\rho_{0},\quad\tilde{v}=b_{0}\tilde{u}\qquad\,\,$
$\displaystyle\mbox{for $\sigma=b_{0}$},$

and define
$\big{(}\tilde{u}(\sigma;s_{0}),\tilde{v}(\sigma;s_{0})\big{)}=(1,0)$ for
$\sigma<s_{0}$. Indeed, we have the following lemma.

Lemma 2.7. Let $1<\gamma<3$ and $p_{0}\in(0,p^{*})$. For $M_{\infty}>K_{1}$,
problem (2.15) has a unique solution
$((\tilde{u}(\sigma;s_{0}),\,\tilde{v}(\sigma;s_{0}))$ containing a supersonic
conical shock-front issuing from the vertex. In addition,

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}(\sigma,\tilde{c}^{2}(\sigma;s_{0}))=(\tan\theta_{0},c_{0}^{2}),$
(2.17)
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}(\tilde{u}(\sigma;s_{0}),\tilde{v}(\sigma;s_{0})=(\cos^{2}\theta_{0},\,\sin\theta_{0}\cos\theta_{0}),$

and

$\lim_{M_{\infty}\rightarrow\infty}\frac{\tilde{u}(\sigma;s_{0})}{\tilde{c}(\sigma;s_{0})}=\frac{\gamma-1-2c_{0}^{2}}{(\gamma-1)c_{0}}>1,\qquad\cos(\theta_{0}\pm\theta_{m}^{0})>0,$
(2.18)

where $\theta_{0}=-\arctan\sqrt{\frac{2c_{0}^{2}}{\gamma-1-2c_{0}^{2}}}$ and
$\theta_{m}^{0}=\lim_{M_{\infty}\rightarrow\infty}\theta_{m}$ for
$\sigma\in[s_{0},b_{0})$.

Proof. Given $p_{0}\in(0,p^{*})$, by the shooting method as in [20], problem
(2.15) has a unique solution
$\big{(}\tilde{u}(\sigma;s_{0}),\tilde{v}(\sigma;s_{0})\big{)}$ with
$\tilde{u}(s_{0};s_{0})>\tilde{c}(s_{0};s_{0})$,
$\tilde{\rho}(b_{0};s_{0})=p_{0}^{\frac{1}{\gamma}}$, and
$\tilde{v}(b_{0};s_{0})=b_{0}\tilde{u}(b_{0};s_{0})$.

We then focus on the asymptotic expansions (2.17). Lemma 2 indicates that

$\displaystyle\frac{1}{1+s_{0}^{2}}<\tilde{u}(\sigma;s_{0})\leq\tilde{u}(s_{0};s_{0}),\qquad\tilde{c}(s_{0};s_{0})\leq\tilde{c}(\sigma;s_{0})<c_{0}<\sqrt{\frac{(\gamma-1)s_{0}^{2}}{2(1+s_{0}^{2})}+\frac{1}{M_{\infty}^{2}}},$

for $\sigma\in[s_{0},b_{0})$. Meanwhile, it follows from (2.10) that
$\tilde{c}(s_{0};s_{0})>c_{d}$. Then, due to Lemma 2, we see that
$u^{\sharp}<\tilde{u}(s_{0};s_{0})<u_{d}$ and $s^{\sharp}<s_{0}<0$. Therefore,
we have

$\displaystyle
c_{d}<\tilde{c}(s_{0};s_{0})\leq\tilde{c}(\sigma;s_{0})<c,\qquad
u_{a}=\frac{1}{1+(s^{\sharp})^{2}}<\frac{1}{1+s_{0}^{2}}<$
$\displaystyle\tilde{u}(\sigma;s_{0})\leq\tilde{u}(s_{0};s_{0})<u_{d}.$

From Lemma 2.1–2.1, we have

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\tilde{c}(\sigma;s_{0})=c_{0}^{2},\qquad\lim_{M_{\infty}\rightarrow\infty}\tilde{u}(\sigma;s_{0})=1-\frac{2c_{0}^{2}}{\gamma-1}.$

Since $\tilde{v}(\sigma;s_{0})<0$, from the Bernoulli laws,

$\displaystyle\frac{\tilde{u}^{2}+\tilde{v}^{2}}{2}+\frac{\tilde{c}^{2}}{\gamma-1}=\frac{1}{2}+\frac{c_{\infty}^{2}}{\gamma-1},\quad\,\frac{(u^{\sharp})^{2}+(v^{\sharp})^{2}}{2}+\frac{c_{0}^{2}}{\gamma-1}=\frac{1}{2}+\frac{c_{\infty}^{2}}{\gamma-1},$

we conclude

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\tilde{v}(\sigma;s_{0})=$
$\displaystyle-\sqrt{\frac{2c_{0}^{2}}{\gamma-1-2c_{0}^{2}}}\,\Big{(}1-\frac{2c_{0}^{2}}{\gamma-1}\Big{)}.$

Again, by Lemma 2, we know that

$\displaystyle\frac{\tilde{v}(s_{0};s_{0})}{\tilde{u}(s_{0};s_{0})}>b_{0}>\sigma\geq
s_{0}>s^{\sharp}.$

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

Furthermore, since

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\big{(}\tilde{u}\sqrt{\tilde{u}^{2}+\tilde{v}^{2}-\tilde{c}^{2}}\big{)}^{2}-(\tilde{v}\tilde{c})^{2}=\lim_{M_{\infty}\rightarrow\infty}(\tilde{u}^{2}-\tilde{c}^{2})(\tilde{u}^{2}+\tilde{v}^{2})>0,$

we conclude

$\displaystyle\cos(\theta_{0}\pm\theta^{0}_{ma})=\lim_{M_{\infty}\rightarrow\infty}\cos(\theta\pm\theta_{m})=\lim_{M_{\infty}\rightarrow\infty}\frac{\tilde{u}\sqrt{\tilde{u}^{2}+\tilde{v}^{2}-\tilde{c}^{2}}\mp\tilde{v}\tilde{c}}{\tilde{u}^{2}+\tilde{v}^{2}}>0.$

This completes the proof. $\square$

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

Lemma 2.8. For $G(s)=\big{(}\bar{u}(s),\bar{v}(s)\big{)}^{\top}$,

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}(\bar{u}(s_{0}),\,\bar{v}(s_{0}))=(\cos^{2}\theta_{0},\,\cos\theta_{0}\sin\theta_{0}),$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}(\bar{u}_{s}(s_{0}),\,\bar{u}_{s}(s_{0}))=(-\sin
2\theta_{0}\cos^{2}\theta_{0},\,\cos 2\theta_{0}\cos^{2}\theta_{0}),$

where $\bar{u}_{s}(s)=\frac{{\rm d}\bar{u}(s)}{{\rm d}s}$ and
$\bar{v}_{s}(s)=\frac{{\rm d}\bar{v}(s)}{{\rm d}s}$.

Now, we introduce the elementary wave curves of system (2.1). We denote by
$W(p_{0},p_{\infty})$ the curve formed by
$\tilde{U}(\sigma;s_{0})=\big{(}\tilde{u}(\sigma;s_{0}),\tilde{v}(\sigma;s_{0})\big{)}^{\top}$
for $s_{0}<\sigma<b_{0}$, where $p_{0}$ is the corresponding pressure of the
state at the endpoint. As in [48] (also cf. [51, 52]), we parameterize the
elementary $i$-wave curves for system (2.1) in a neighborhood of
$W(p_{0},p_{\infty})$:

$O_{r}\big{(}W(p_{0},p_{\infty})\big{)}=\bigcup\limits_{s_{0}<\sigma<b_{0}}\\{U:\
|U-\tilde{U}(\sigma;s_{0})|<r\\}\qquad\mbox{for some $r>0$}$

by

$\displaystyle\alpha_{i}\mapsto\Phi_{i}(\alpha_{i};U)$

with $\Phi_{i}\in\text{C}^{2}$ and

$\displaystyle\left.\frac{\partial\Phi_{i}}{\partial\alpha_{i}}\right|_{\alpha_{i}=0}=r_{i}(U)\qquad\mbox{for
$U\in O_{r}(W(p_{0},p_{\infty}))$},\,\,i=1,2.$

In the sequel, define

$\displaystyle\Phi(\alpha_{1},\alpha_{2};U)=\Phi_{2}\big{(}\alpha_{2};\Phi_{1}(\alpha_{1};U)\big{)}.$

Denote $\tilde{U}(\sigma;\sigma_{0},U_{l})$ the solution to the ODE system
(2.3) with initial data

$\displaystyle\tilde{U}|_{\sigma=\sigma_{0}}=U_{l}$

for $U_{l}\in O_{r}\big{(}W(p_{0},p_{\infty})\big{)}$. Then, as in [48], we
have

Lemma 2.9. For $p_{0}\in(0,p^{*})$,

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\left.\frac{{\rm
d}\tilde{U}(\sigma;\sigma_{0})}{{\rm
d}\sigma}\right|_{\\{\sigma=\sigma_{0},\,U_{l}\in
W(p_{0},p_{\infty})\\}}=\big{(}\sin\theta_{0}\cos^{3}\theta_{0},-\cos^{4}\theta_{0}\big{)}^{\top}.$

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

Lemma 2.10. For $U_{l}\in W(p_{0},p_{\infty})$,

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\big{(}r_{1}(U_{l}),r_{2}(U_{l})\big{)}=\dfrac{4\cos^{2}(\theta_{0}+\theta_{m}^{0})\cos^{2}(\theta_{0}-\theta_{m}^{0})\cos^{2}\theta_{0}\cos^{2}\theta^{0}_{m}\sin(2\theta_{m}^{0})}{(\gamma+1)^{2}},$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\big{(}G^{\prime}(s_{0}),r_{1}\big{(}G(s_{0})\big{)}\big{)}=-\dfrac{2\cos^{2}(\theta_{0}-\theta_{m}^{0})\cos\theta^{0}_{m}\cos^{3}\theta_{0}\sin(\theta_{0}+\theta_{m}^{0})}{\gamma+1},$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\big{(}r_{2}\big{(}G(s_{0})\big{)},G^{\prime}(s_{0})\big{)}=\dfrac{2\cos^{2}(\theta_{0}+\theta_{m}^{0})\cos\theta^{0}_{m}\cos^{3}\theta_{0}\sin(\theta_{0}-\theta_{m}^{0})}{\gamma+1},$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\Big{(}\frac{{\rm
d}\tilde{U}(\sigma;\sigma_{0},G(s_{0}))}{{\rm
d}\sigma},G^{\prime}(s_{0})\Big{)}=-\cos^{5}\theta_{0}\sin\theta_{0},$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\Big{(}r_{2}(U_{l}),\frac{{\rm
d}\tilde{U}(\sigma;\sigma_{0},U_{l})}{{\rm
d}\sigma}\Big{)}=\frac{2}{\gamma+1}\cos^{4}\theta_{0}\cos^{2}(\theta_{0}+\theta_{m}^{0})\cos\theta_{m}^{0}\sin\theta_{m}^{0},$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\Big{(}r_{1}(U_{l}),\frac{{\rm
d}\tilde{U}(\sigma;\sigma_{0},U_{l})}{{\rm
d}\sigma}\Big{)}=-\frac{2}{\gamma+1}\cos^{4}\theta_{0}\cos^{2}(\theta_{0}-\theta_{m}^{0})\cos\theta_{m}^{0}\sin\theta_{m}^{0}.$

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

Proposition 2.1. For $M_{\infty}$ sufficiently large, there exists
$\varepsilon_{1}>0$ such that, for any $U_{r}$ and $U_{l}$ lie in
$O_{\varepsilon_{1}}\big{(}W(p_{0},p_{\infty})\big{)}$, the Riemann problem
(2.1) with initial data

$U|_{x=\bar{x}}=\left\\{\begin{aligned} &U_{r}&\,\,\,\text{for }y>\bar{y},\\\
&U_{l}&\,\,\,\text{for }y<\bar{y},\end{aligned}\right.$ (2.19)

admits a unique admissible solution consisting of at most two elementary waves
$\alpha_{1}$ for the $1$-characteristic field and $\alpha_{2}$ for the
$2$-characteristic field. Moreover, states $U_{l}$ and $U_{r}$ are connected
by

$\displaystyle U_{r}=\Phi(\alpha_{1},\alpha_{2};U_{l}).$

Proposition 2.2. For $M_{\infty}$ sufficiently large, there exists
$\varepsilon_{2}>0$ such that, for any $U_{l}\in
O_{\varepsilon_{2}}\big{(}W(p_{0},p_{\infty})\big{)}$ and $p_{1}$, $p_{2}\in
O_{\varepsilon_{2}}(p_{0})$, there is $\delta_{1}$ solving the equation:

$\dfrac{1}{2}|\Phi(\delta_{1},0;U_{l})|^{2}+\dfrac{\gamma}{\gamma-1}p_{2}^{\frac{\gamma-1}{\gamma}}=\dfrac{1}{2}|U_{l}|^{2}+\dfrac{\gamma}{\gamma-1}p_{1}^{\frac{\gamma-1}{\gamma}}.$
(2.20)

Proof. From (2.20), we have

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\frac{1}{2}\,\left.\frac{\partial|\Phi(\delta_{1},0;U_{l})|^{2}}{\partial\delta_{1}}\right|_{\delta_{1}=0}=U_{l}\cdot
r_{1}(U_{l})\neq 0.$

By the implicit function theorem, there exists $\delta_{1}$ such that (2.20)
holds, provided $\varepsilon_{2}$ sufficiently small. $\square$

Proposition 2.3. For $M_{\infty}$ sufficiently large, there exists
$\varepsilon_{3}>0$ such that, for any $U_{l}=U_{\infty}$ and $U_{r}\in
O_{\varepsilon_{3}}(W(p_{0},p_{\infty}))\cap O_{\varepsilon_{3}}(G(s_{0}))$,
the Riemann problem (2.1) with initial data (2.19) admits a unique admissible
solution that contains a strong $1$-shock $s_{1}$ and a $2$-weak wave
$\beta_{2}$ of the 2-characteristic field. Moreover, states $U_{l}$ and
$U_{r}$ are connected by

$U_{r}=\Phi_{2}(\beta_{2};G(s_{1};U_{l})).$ (2.21)

Proof. It follows from (2.21) and Lemma 2.1 that

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\Big{(}\left.\frac{\partial\Phi_{2}(\beta_{2};G(s_{1};U_{l}))}{\partial(s_{1},\beta_{2})}\right|_{\\{s_{1}=s_{0},\beta_{2}=0\\}}\Big{)}=-\lim_{M_{\infty}\rightarrow\infty}\det\big{(}r_{2}(G(s_{0}),G^{\prime}(s_{0}))\big{)}\neq
0.$

The existence of the solution of this Riemann problem is ensured by the
implicit function theorem for $\varepsilon_{3}$ sufficiently small. $\square$

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

Lemma 2.11. Let $U_{l}\in W(p_{0},p_{\infty})$, $\alpha$, $\beta$, and
$\delta$ satisfy

$\displaystyle\Phi(\delta_{1},\delta_{2};U_{l})=\Phi(\beta_{1},\beta_{2};\Phi(\alpha_{1},\alpha_{2};U_{l})).$

Then

$\displaystyle\delta=\alpha+\beta+O(1)Q^{0}(\alpha,\beta),$

where $Q^{0}(\alpha,\beta)=\sum\\{|\alpha_{i}||\beta_{i}|:\text{$\alpha_{i}$
and $\beta_{j}$ approach}\\}$, and $O(1)$ depends continuously on
$M_{\infty}<\infty$.

##  3\. Approximate Solutions

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

Given $\epsilon>0$ and $\Delta x>0$, there exist piece-wise constant functions
$p^{b}_{\Delta x}$ such that

$\displaystyle\text{T.V.}~{}p^{b}_{\Delta
x}(\cdot)\leq\text{T.V.}~{}p^{b}(\cdot),\qquad\|p^{b}_{\Delta
x}-p^{b}\|_{\textbf{L}^{\infty}}\leq\epsilon,$

where

$\displaystyle p^{b}_{\Delta x}(x)=\begin{cases}p^{b}_{\Delta
x,0}=p_{0}\,\,\,&\text{ for $x\in[0,x_{0})$},\\\ p^{b}_{\Delta
x,h+1}\,\,\,&\text{ for $x\in[x_{h},x_{h+1})$ and
$h\in\mathbb{N}$},\end{cases}$

with $p^{b}_{\Delta x,h+1}$ being constants on the corresponding intervals and
$x_{h}=x_{0}+h\Delta x$ for $h\in\mathbb{N}$. Then, from Lemma 2.1, for
$p^{b}_{\Delta x,0}=p_{0}$, there exists
$\big{(}\tilde{u}(\sigma;s_{0}),\tilde{v}(\sigma;s_{0})\big{)}$ such that
$\tilde{p}(b_{0};s_{0})=p_{0}$ and
$\tilde{v}(b_{0};s_{0})=b_{0}\tilde{u}(b_{0};s_{0})$.

We now define the difference scheme. Choose
$\vartheta=(\vartheta_{0},\vartheta_{1},\vartheta_{2},\dots,\vartheta_{h},\dots)$
randomly in $[0,1)$. For $0<x<x_{0}$, let

$\displaystyle b_{\Delta x,\vartheta}(x)=b_{0}x,\qquad\,\,\chi_{\Delta
x,\vartheta}(x)=s_{0}x.$

We denote $\Gamma_{\Delta x,\vartheta,0}=\big{\\{}(x,b_{\Delta
x,\vartheta}(x))\,:\,0\leq x<x_{0}\big{\\}}$, $S_{\Delta
x,\vartheta,0}=\big{\\{}(x,\chi_{\Delta x,\vartheta}(x))\,:\,0\leq
x<x_{0}\big{\\}}$, and $\Omega_{\Delta
x,\vartheta,0}=\big{\\{}(x,y)\,:\,y<b_{\Delta x,\vartheta}(x),0\leq
x<x_{0}\big{\\}}$. In region $\Omega_{\Delta x,\vartheta,0}$, we then define

$\displaystyle U_{\Delta x,\vartheta}(x,y)=\begin{cases}(u_{\Delta
x,\vartheta}(x,y),v_{\Delta
x,\vartheta}(x,y))^{\top}\triangleq(\tilde{u}(\sigma;s_{0}),\tilde{v}(\sigma;s_{0}))^{\top}\,\,\,&\text{for
$\frac{y}{x}=\sigma\in(s_{0},b_{0})$},\\\ U_{\infty}\quad&\text{for
$\frac{y}{x}=\sigma<s_{0}$},\end{cases}$

and, on boundary $\Gamma_{\Delta x,\vartheta,0}$, we set

$\displaystyle U_{\Delta x,\vartheta}(x,b_{\Delta x,\vartheta}(x))=U_{\Delta
x,\vartheta}^{b}(x)=(u_{\Delta x,\vartheta}^{b}(x),v_{\Delta
x,\vartheta}^{b}(x))^{\top}\triangleq(\tilde{u}(b_{0};s_{0}),\tilde{v}(b_{0};s_{0}))^{\top}.$

On $x=x_{h}$ for $h\in\mathbb{N}$, the grid points are defined to be the
intersections of line $x=x_{h}$ with the self-similar rays

$\displaystyle y=\left(b_{0}+n\Delta\sigma\right)x\qquad\mbox{for
$n\in\mathbb{Z}$}.$

Here $\Delta\sigma>0$ is chosen so that $\Delta\sigma>\dfrac{4\Delta
x}{x_{0}}\max_{i=1,2}\\{|\lambda_{i}(G(s_{0}))|\\}$, and hence the numerical
grids satisfies the usual Courant-Friedrichs-Lewy condition. Then we define
the approximate solution $U_{\Delta x,\vartheta}$ to be a piece-wise smooth
solution to the self-similar system (2.4), the approximate solution $U_{\Delta
x,\vartheta}^{b}$ on the boundary, the approximate boundary $\Gamma_{\Delta
x,\vartheta}=\big{\\{}(x,y)\,:\,y=b_{\Delta x,\vartheta}(x)\big{\\}}$, and the
numerical grids inductively in $h$, $h=0,1,2,\cdots$.

Suppose that the approximate solution has been defined on $x<x_{h}$. The grid
points on $x=x_{h}$ are denoted by $y_{n}(h)$ for $n\in\mathbb{Z}$. Set

$\displaystyle
r_{h,n}=y_{n}(h)+\vartheta_{h}\big{(}y_{n+1}(h)-y_{n}(h)\big{)}\qquad\,\,\mbox{for
$n\in\mathbb{Z}$}.$

Then the approximate solution $U_{\Delta x,\vartheta}(x_{h},y)$ for
$y\in(y_{n}(h),y_{n+1}(h))$ is defined to be the solution $U_{self,\Delta
x,\vartheta}(\sigma(x,y))$ of (2.3) with the self-similar variable
$\sigma(x,y)=\frac{y}{x_{h}}$ and with the initial data:

$\displaystyle\sigma=\frac{r_{h,n}}{x_{h}}:\quad U_{self,\Delta
x,\vartheta}=U_{\Delta x,\vartheta}(x_{h},r_{h,n})\triangleq U_{\Delta
x,\vartheta}(x_{h}-,r_{h,n}+)\qquad\mbox{for $n\in\mathbb{Z}$}.$

For the discontinuities at the grid points $\big{(}x_{h},y_{n}(h)\big{)}$ for
$n\in\mathbb{Z}$, we solve the Riemann problems for (2.1) with the Riemann
data:

$U|_{x=x_{h}}=\left\\{\begin{aligned} &U_{\Delta
x,\vartheta}(x_{h},y_{n}(h)-)\qquad\mbox{for $y<y_{n}(h)$},\\\ &U_{\Delta
x,\vartheta}(x_{h},y_{n}(h)+)\qquad\mbox{for
$y>y_{n}(h)$},\end{aligned}\right.$ (3.1)

and the solution consisting of rarefaction waves and shock waves has form
$U_{Rie}(\eta)$ with $\eta=\dfrac{y-y_{n}(h)}{x-x_{h}}$. Setting
$\sigma_{h,n+\frac{1}{2}}\triangleq\dfrac{1}{2x_{h}}\big{(}y_{n+1}(h+1)+y_{n}(h+1)\big{)}$
for $n\in\mathbb{Z}$, then, in the region:

$\displaystyle\Omega_{h+1,n}=\big{\\{}(x,y)\,:\,x_{h}<x<x_{h+1},\sigma_{h,n+\frac{1}{2}}>\sigma>\sigma_{h,n-\frac{1}{2}}\big{\\}},$

along the ray

$\displaystyle\big{\\{}(x,y)\,:\,\frac{y-y_{n}(h)}{x-x_{h}}=\eta,\,x_{h}<x<x_{h+1}\big{\\}},$

the approximate solution $U_{\Delta x,\vartheta}(x,y)$ is defined to be the
solution: $U_{self,\Delta x,\vartheta}\big{(}\sigma(x,y)\big{)}$ of (2.3) with
the self-similar variable $\sigma(x,y)=\frac{y}{x}$ and with the initial data:

$\displaystyle\sigma=\frac{y_{n}}{x_{h}}:\quad U_{self,\Delta
x,\vartheta}=U_{Rie}(\eta).$

The approximate boundary $\Gamma_{\Delta
x,\vartheta}=\big{\\{}(x,y)\,:\,y=b_{\Delta x,\vartheta}(x)\big{\\}}$ is
traced continuously; see [38, 48, 10]. For $x\in(0,x_{0})$, let $b_{\Delta
x,\vartheta}(x)=b_{0}x$. Suppose that the approximate solution is constructed
for $x<x_{h}$ and that $y_{n_{b,h}}<b_{\Delta
x,\vartheta}(x_{h}-)<y_{n_{b,h}+1}$. We call interval
$y_{n_{b,h}-1}<y<y_{n_{b,h}+1}$ the boundary region at $x=x_{h}$. In this
boundary region, we first solve the self-similar problem (2.3) with the
initial data:

$\displaystyle\sigma=\frac{r_{h,n_{b}-1}}{x_{h}}:\quad U_{self}=U_{\Delta
x,\vartheta}(x_{h}-,r_{h,n_{b}-1}+),$

and with the self-similar variable $\sigma(x_{h},y)=\frac{y}{x_{h}}$. We
denote the solution by $U_{self}\big{(}\sigma(x_{h},y)\big{)}$. Given
$p^{b}_{\Delta x,h+1}$, by Proposition 2.1, there is $\beta_{1}$ such that

$\displaystyle\frac{1}{2}\big{|}\Phi(\beta_{1},0;U_{self}(\sigma\big{(}x_{h},b_{\Delta
x,\vartheta}(x_{h}))))\big{|}^{2}+\frac{\gamma}{\gamma-1}(p^{b}_{\Delta
x,h+1})^{\frac{\gamma-1}{\gamma}}=\frac{1}{2}+\frac{\gamma}{\gamma-1}p_{\infty}^{\frac{\gamma-1}{\gamma}}.$

Then we define

$\displaystyle U_{\Delta
x,\vartheta}^{b}(x_{h})\triangleq\Phi(\beta_{1},0;U_{self}(\sigma(x_{h},b_{\Delta
x,\vartheta}(x_{h})))),$

and

$b_{\Delta x,\vartheta}(x)=b_{\Delta x,\vartheta}(x_{h}-)+\frac{v^{b}_{\Delta
x,\vartheta}(x_{h})}{u^{b}_{\Delta
x,\vartheta}(x_{h})}\,(x-x_{h})\qquad\mbox{for $x\in[x_{h},x_{h+1})$}.$ (3.2)

Next, solve again the self-similar problem (2.3) with initial data
$U_{-}(\sigma(x_{h},b_{\Delta x,\vartheta}(x_{h})))=U_{\Delta
x,\vartheta}^{b}(x_{h})$ and with the self-similar variable
$\sigma(x_{h},y)=\frac{y}{x_{h}}$. Denote the solution by
$U_{-}(\sigma(x_{h},y))$. We define the approximate solution in the boundary
region as

$\displaystyle U_{\Delta
x,\vartheta}(x_{h},y)=U_{-}(\sigma(x_{h},y))\qquad\mbox{for $x_{h}\leq
x<x_{h+1}$}.$

The discontinuities at $(x_{h},y_{n_{b,h}-1})$ are resolved by the same
methods as before.

The leading strong conical shock $S_{\Delta
x,\vartheta}=\big{\\{}(x,y)\,:\,y=\chi_{\Delta x,\vartheta}(x)\big{\\}}$ next
to the uniform upstream flow is also traced continuously; see [38, 48, 10].
For $x\in(0,x_{0})$, let $\chi_{\Delta x,\vartheta}(x)=s_{0}x$. Suppose that
the approximate solution is constructed for $x<x_{h}$ and that
$y_{n_{\chi,h}-1}<\chi_{\Delta x,\vartheta}(x_{h}-)<y_{n_{\chi,h}}$. We call
interval $y_{n_{\chi,h}-1}<y<y_{n_{\chi,h}+1}$ the front region at $x=x_{h}$.
In this front region, we first solve the self-similar problem (2.3) with the
initial data

$\displaystyle\sigma=\frac{r_{h,n_{\chi}}}{x_{h}}:\quad U_{self}=U_{\Delta
x,\vartheta}(x_{h}-,r_{h,n_{\chi}}+),$

and with the self-similar variable $\sigma(x_{h},y)=\frac{y}{x_{h}}$. Denote
the solution by $U_{self}(\sigma(x_{h},y))$. Then we solve the Riemann problem
(2.1) with the initial data

$U(x_{h},y)=\left\\{\begin{aligned} &U_{\infty},\,\,\,&&y<\chi_{\Delta
x,\vartheta}(x_{h}-),\\\ &U_{self}(\sigma(x_{h},\chi_{\Delta
x,\vartheta}(x_{h}))),\,\,\,&&\chi_{\Delta
x,\vartheta}(x_{h}-)<y<y_{n_{\chi,h}+1}.\end{aligned}\right.$ (3.3)

The solution $U(x,y)$ contains a weak 2-wave $\beta_{2}$ and a relatively
strong 1-shock wave $s_{\Delta x,\vartheta}(h+1)$ such that

$\displaystyle U_{self}(\sigma(x_{h},\chi_{\Delta
x,\vartheta}(x_{h})))=\Phi(0,\beta_{2};G(s_{\Delta
x,\vartheta}(h+1);U_{\infty})).$

Then, let

$\chi_{\Delta x,\vartheta}(x)=\chi_{\Delta x,\vartheta}(x_{h}-)+s_{\Delta
x,\vartheta}(h+1)(x-x_{h})\qquad\mbox{for $x\in[x_{h},x_{h+1})$}.$ (3.4)

Next, solve again the self-similar problem (2.3) with initial data
$U_{+}(\sigma(x_{h},\chi_{\Delta x,\vartheta}(x_{h})))=G(s_{\Delta
x,\vartheta}(h+1);U_{\infty})$ and with the self-similar variable
$\sigma(x_{h},y)=\frac{y}{x_{h}}$. Denote the solution by
$U_{+}(\sigma(x_{h},y))$. We define the approximate solution in the front
region as

$\displaystyle U_{\Delta x,\vartheta}(x_{h},y)=\left\\{\begin{aligned}
&U_{\infty},\,\,\,&&y<\chi_{\Delta x,\vartheta}(x_{h}),\\\
&U_{+}(\sigma(x_{h},y)),\,\,\,&&\chi_{\Delta
x,\vartheta}(x_{h})<y<y_{n_{\chi,h}+1}.\end{aligned}\right.$

The discontinuities at $(x_{h},y_{n_{\chi,h}})$ are resolved by the same
methods as before.

##  4\. Riemann-Type Problems and Interaction Estimates

Let $\Omega_{\Delta x,\vartheta,h}=\\{(x,y)\,:\,y<b_{\Delta
x,\vartheta},x_{h-1}\leq x<x_{h}\\}$ and $h\in\mathbb{N}_{+}$. In order to
define the approximate solutions in $\Omega_{\Delta
x,\vartheta}\triangleq\bigcup_{k=0}^{\infty}\Omega_{\Delta x,\vartheta,k}$,
the approximate boundary $\Gamma_{\Delta
x,\vartheta}\triangleq\bigcup_{k=0}^{\infty}\Gamma_{\Delta x,\vartheta,k}$,
and the approximate leading shock $S_{\Delta
x,\vartheta}\triangleq\bigcup_{k=0}^{\infty}S_{\Delta x,\vartheta,k}$, we need
a uniform bound of them to ensure that all the Riemann problems and the
differential equations (2.3) are solvable. To achieve this, the following
formulas are used:

  1. (i)

If $f\in\text{C}^{1}(\mathbb{R})$, then

$f(t)-f(0)=t\int_{0}^{1}f_{t}(\mu t){\rm d}\mu\qquad\,\,\mbox{for
$t\in\mathbb{R}$}.$ (4.1)

  2. (ii)

If $f\in\text{C}^{2}(\mathbb{R})$, then

$f(s,t)-f(s,0)-f(0,t)+f(0,0)=st\int_{0}^{1}\int_{0}^{1}f_{st}(\mu s,\lambda
t){\rm d}\mu{\rm d}\lambda\qquad\,\,\mbox{for $(s,t)\in\mathbb{R}^{2}$}.$
(4.2)

From now on, we use Greek letters $\alpha$, $\beta$, $\gamma$, and $\delta$ to
denote the elementary waves in the approximate solution, and $\alpha_{i}$,
$\beta_{i}$, $\gamma_{i}$, and $\delta_{i}$ stand for the corresponding $i$-th
components for $i=1,2$. As in [14, 44, 48, 52], a curve $I$ is called a mesh
curve provided that $I$ is a space-like curve and consists of the line
segments joining the random points one by one in turn. $I$ divides region
$\Omega_{\Delta x,\vartheta}$ into two parts: $I^{-}$ and $I^{+}$, where
$I^{-}$ denotes the part containing line $x=x_{0}$. For any two mesh curves
$I$ and $J$, we use $J>I$ to represent that every mesh point of curve $J$ is
either on $I$ or contained in $I^{+}$. We say $J$ is an immediate successor to
$I$ if $J>I$ and every mesh point of $J$ except one is on $I$ in general but
three when these points are near the approximate boundary or the approximate
shock.

Assume now that $U_{\Delta x,\vartheta}$ has been defined in
$\bigcup_{k=0}^{h}\Omega_{\Delta x,\vartheta,k}$ and the following conditions
are satisfied:

  * $H_{1}(h)$:

$\\{S_{\Delta x,\vartheta,k}\\}_{k=0}^{h}$ forms an approximate strong shock
$S_{\Delta x,\vartheta}|_{0\leq x<x_{h}}$, and $\\{\Gamma_{\Delta
x,\vartheta,k}\\}_{k=0}^{h}$ forms an approximate boundary $\Gamma_{\Delta
x,\vartheta}|_{0\leq x<x_{h}}$, both of which emanate from the origin.

  * $H_{2}(h)$:

In each $\Omega_{\Delta x,\vartheta,k}$ for $0\leq k\leq h$, the strong
1-shock $S_{\Delta x,\vartheta,k}$ divides $\Omega_{\Delta x,\vartheta,k}$
into two parts: $\Omega_{\Delta x,\vartheta,k}^{-}$ and $\Omega_{\Delta
x,\vartheta,k}^{+}$, where $\Omega_{\Delta x,\vartheta,k}^{+}$ is the part
between $S_{\Delta x,\vartheta,k}$ and $\Gamma_{\Delta x,\vartheta,k}$;

  * $H_{3}(h)$:

$U_{\Delta x,\vartheta}|_{\Omega_{\Delta x,\vartheta,k}^{-}}=U_{\infty}$,
$\,\,U_{\Delta x,\vartheta}|_{\Omega_{\Delta x,\vartheta,k}^{+}}\in
O_{\varepsilon_{0}}(G(s_{0}))\cap O_{\varepsilon_{0}}(W(p_{0},p_{\infty}))$,
and

$U_{\Delta x,\vartheta}(x,b_{\Delta x,\vartheta}(x)-)=U_{\Delta
x,\vartheta,k}^{b}\in O_{\varepsilon_{0}}(G(s_{0}))\cap
O_{\varepsilon_{0}}(W(p_{0},p_{\infty}))$

for $x_{k}\leq x<x_{k+1}$, $0\leq k\leq h$, and
$0<\varepsilon_{0}<\min\\{\varepsilon_{j},j=1,2,3\\}$, where $\varepsilon_{j}$
are introduced in Propositions 2.1–2.1 for $j=1,2,3$.

We prove that $U_{\Delta x,\vartheta}$ can be defined in $\Omega_{\Delta
x,\vartheta,h+1}$ satisfying $H_{1}(h+1)$ – $H_{3}(h+1)$. As in [24] (see also
[14, 44, 10]), we consider a pair of the mesh curves $(I,J)$ lying in
$\\{x_{h-1}<x<x_{h+1}\\}\cap\Omega_{\Delta x,\vartheta}$ with $J$ being an
immediate successor of $I$.

Now, let $\Lambda$ be the region between $I$ and $J$, and let

$\displaystyle U_{\Delta x,\vartheta}\in O_{\varepsilon_{0}}(G(s_{0}))\cap
O_{\varepsilon_{0}}(W(p_{0},p_{\infty})).$

Case 1. $\Lambda$ is between $\Gamma_{\Delta x,\vartheta}$ and $S_{\Delta
x,\vartheta}$. In this case, we consider the interactions between weak waves.
From the construction of the approximate solutions, the waves entering
$\Lambda$ issuing from $(x_{h-1},y_{n-1}(h-1))$ and from
$(x_{h-1},y_{n}(h-1))$ are denoted by $\alpha=(\alpha_{1},\alpha_{2})$ and
$\beta=(\beta_{1},\beta_{2})$, respectively. We denote

$\displaystyle\sigma_{0}$ $\displaystyle=\frac{r_{h-1,n}}{x_{h-1}},\ $
$\displaystyle\bar{\sigma}_{0}$ $\displaystyle=\frac{r_{h-1,n-1}}{x_{h-1}},\ $
$\displaystyle\hat{\sigma}_{0}$ $\displaystyle=\frac{r_{h-1,n-2}}{x_{h-1}},$
$\displaystyle\sigma_{1}$
$\displaystyle=\frac{y_{n}(h-1)}{x_{h-1}}=\frac{y_{n}(h)}{x_{h}},\ $
$\displaystyle\sigma_{2}$
$\displaystyle=\frac{y_{n-1}(h-1)}{x_{h-1}}=\frac{y_{n-1}(h)}{x_{h}},$

and

$\displaystyle U_{1}=U_{\Delta x,\vartheta}(x_{h-1}-,r_{h-1,n}+),\
U_{2}=U_{\Delta x,\vartheta}(x_{h-1}-,r_{h-1,n-1}+),\ U_{3}=U_{\Delta
x,\vartheta}(x_{h-1}-,r_{h-1,n-2}+).$

Fig. 4.1. Interaction between weak waves

Case 1.1. Let $\delta=(\delta_{1},\delta_{2})$ be the waves issuing from
$(x_{h},y_{n-1}(h))$; see Fig. 4.1. Then we need to solve the following
equations of $\delta=(\delta_{1},\delta_{2})$:

$\tilde{U}(\sigma_{1};\sigma_{2},\Phi(\delta_{1},\delta_{2};U_{l}))=\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(\alpha_{1},\alpha_{2};U_{l}))),$
(4.3)

where $U_{l}=\tilde{U}(\sigma_{2};\hat{\sigma}_{0},U_{3})$.

Lemma 4.1. Equation (4.3) has a unique solution
$\delta=(\delta_{1},\delta_{2})$ such that

$\displaystyle\delta_{1}=\alpha_{1}+\beta_{1}+O(1)Q(\Lambda),\qquad\delta_{2}=\alpha_{2}+O(1)Q(\Lambda),$

where

$\displaystyle Q(\Lambda)=Q^{0}(\Lambda)+Q^{1}(\Lambda)$

with

$\displaystyle
Q^{0}(\Lambda)=\sum\big{\\{}|\alpha_{j}||\beta_{k}|\,:\,\alpha_{j}\text{ and
}\beta_{k}\text{ approach}\big{\\}},\qquad
Q^{1}(\Lambda)=|\beta_{1}||\Delta\sigma|,$

and $\Delta\sigma=\sigma_{1}-\sigma_{2}$, where $O(1)$ depends continuously on
$M_{\infty}$ but independent of $(\alpha,\beta,\Delta\sigma)$.

Proof. Lemma 2.1 yields

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\det\bigg{(}\left.\frac{\partial\Phi(\delta_{1},\delta_{2};U_{l})}{\partial(\delta_{1},\delta_{2})}\right|_{\\{\delta_{1}=\delta_{2}=0,\,U_{l}\in
W(p_{0,U_{\infty}})\\}}\bigg{)}$
$\displaystyle=\dfrac{4\cos^{2}(\theta_{0}+\theta_{m}^{0})\cos^{2}(\theta_{0}-\theta_{m}^{0})\cos^{2}\theta_{0}\cos^{2}\theta^{0}_{m}\sin(2\theta_{m}^{0})}{(\gamma+1)^{2}}.$

Then, by the implicit function theorem, system (4.3) has a unique C2–solution:

$\displaystyle\delta=\delta(\alpha,\beta,\Delta\sigma;U_{l})$

in a neighborhood of $(\alpha,\beta,\Delta\sigma,U_{l})=(0,0,0,G(s_{0}))$. Due
to (4.2), we have

$\displaystyle\delta_{i}(\alpha,\beta,\Delta\sigma;U_{l})$
$\displaystyle=\delta_{i}(\alpha,0,\Delta\sigma;U_{l})+\delta_{i}(\alpha,\beta,0;U_{l})-\delta_{i}(\alpha,0,0;U_{l})+O(1)|\beta||\Delta\sigma|$
$\displaystyle=\alpha_{i}+\beta_{i}+O(1)Q^{0}(\Lambda)+O(1)|\beta||\Delta\sigma|\qquad\mbox{for
$i=1,2$},$

where $\beta_{2}=0$. Then the proof is complete. $\square$

Case 1.2. Let $\delta=(\delta_{1},\delta_{2})$ be the waves issuing from
$(x_{h},y_{n}(h))$; see Fig. 4.1. Then we need to solve the following
equations of $\delta=(\delta_{1},\delta_{2})$:

$\Phi(\delta_{1},\delta_{2};\tilde{U}(\sigma_{1};\sigma_{2},U_{l}))=\Phi(\beta_{1},\beta_{2};\tilde{U}(\sigma_{1},\sigma_{2};\Phi(0,\alpha_{2};U_{l}))),$
(4.4)

where $U_{l}$ satisfies
$\tilde{U}(\bar{\sigma}_{0};\sigma_{1},\Phi(0,\alpha_{2};U_{l}))=U_{2}$.
Similarly, we have the following lemma.

Lemma 4.2. Equation (4.4) has a unique solution
$\delta=(\delta_{1},\delta_{2})$ such that

$\displaystyle\delta_{1}=\beta_{1}+O(1)Q(\Lambda),\qquad\delta_{2}=\alpha_{2}+\beta_{2}+O(1)Q(\Lambda),$

where

$\displaystyle Q(\Lambda)=Q^{0}(\Lambda)+Q^{1}(\Lambda)$

with

$\displaystyle
Q^{0}(\Lambda)=\sum\\{|\alpha_{j}||\beta_{k}|\,:\,\alpha_{j}\text{ and
}\beta_{k}\text{ approach}\\},\qquad Q^{1}(\Lambda)=|\alpha||\Delta\sigma|,$

and $\Delta\sigma=\sigma_{1}-\sigma_{2}$. Here $O(1)$ depends continuously on
$M_{\infty}$ but independent of $(\alpha,\beta,\Delta\sigma)$.

Case 2. $\Lambda_{b}$ covers the part of $\Gamma_{\Delta x,\vartheta}$ but
none of $S_{\Delta x,\vartheta}$. We take three diamonds at the same time, as
shown in Fig. 4.2. Let $\Delta_{h,n_{b,h}-1}$, $\Delta_{h,n_{b,h}}$, and
$\Delta_{h,n_{b,h}+1}$ denote the diamonds centering in
$(x_{h},y_{n_{b,h}-1})$, $(x_{h},y_{n_{b,h}})$, and $(x_{h},y_{n_{b,h}+1})$,
respectively, and denote
$\Lambda_{b}=\Delta_{h,n_{b,h}-1}\cup\Delta_{h,n_{b,h}}\cup\Delta_{h,n_{b,h}+1}$.
Let $\alpha$ and $\gamma$ be the weak waves issuing from
$(x_{h-1},y_{n_{b,h-1}-1})$ and $(x_{h-1},y_{n_{b,h-1}-2})$ respectively, and
entering $\Lambda_{b}$. We divide $\alpha=(\alpha_{1},\alpha_{2})$ into parts
$\alpha_{l}=(\alpha_{l,1},0)$ and $\alpha_{r}=(\alpha_{r,1},\alpha_{r,2})$,
where $\alpha_{l}$ and $\alpha_{r}$ entering $\Delta_{h,n_{b,h}-1}$ and
$\Delta_{h,n_{b,h}}$, respectively. Moreover, let
$\gamma=(\gamma_{1},\gamma_{2})$ issuing from $(x_{h-1},y_{n_{b,h}-2})$, and
let $\delta$ be the outgoing wave issuing from $(x_{h},y_{n_{b,h}-1})$. For
simplicity of notation, we denote

$\displaystyle\sigma_{\alpha}=\sigma(x_{h-1},y_{n_{b,h-1}-1}),$
$\displaystyle\sigma_{b}(h-1)=\sigma(x_{h-1},b_{\Delta
x,\vartheta}(x_{h-1})),$ $\displaystyle\sigma_{b}(h)=\sigma(x_{h},b_{\Delta
x,\vartheta}(x_{h})),$
$\displaystyle\Delta\sigma_{\alpha}=\sigma_{b}(h-1)-\sigma_{\alpha},$
$\displaystyle\Delta\bar{\sigma}_{\alpha}=\sigma_{b}(h)-\sigma_{\alpha},$
$\displaystyle\Delta\sigma_{b_{h}}=\sigma_{b}(h)-\sigma_{b}(h-1),$
$\displaystyle\sigma_{\gamma}=\sigma(x_{h-1},y_{n_{b,h-1}-2}),$
$\displaystyle\Delta\sigma_{\gamma}=\sigma_{\alpha}-\sigma_{\gamma},$
$\displaystyle\sigma_{0}=\sigma(x_{h-1},r_{h-1,n_{b}-2}),$

and $U_{1}=U_{\Delta x,\vartheta}(x_{h-1}-,r_{h-1,n_{b}-2}+)$. Let
$U_{l}=\Phi(\alpha_{l,1},0;\tilde{U}(\sigma_{\alpha};\sigma_{0},U_{1}))$.

Fig. 4.2. Reflection at the boundary

To gain the estimates of $\delta$, we need to deal with the equation:

$\displaystyle\dfrac{1}{2}|\Phi(\beta_{1},0;\tilde{U}(\sigma_{b}(h);\sigma_{\alpha},U_{l}))|^{2}+\dfrac{\gamma}{\gamma-1}(p^{b}_{\Delta
x,h+1})^{\frac{\gamma-1}{\gamma}}$
$\displaystyle=\dfrac{1}{2}|\tilde{U}(\sigma_{b}(h-1);\sigma_{\alpha},\Phi(\alpha_{r,1},\alpha_{r,2};U_{l}))|^{2}+\dfrac{\gamma}{\gamma-1}(p^{b}_{\Delta
x,h})^{\frac{\gamma-1}{\gamma}},$ (4.5)

and then we obtain the following lemma.

Lemma 4.3. Equation (4.5) has a unique solution
$\beta_{1}=\beta_{1}(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\omega_{h+1};U_{l})\in\text{C}^{2}$
in a neighborhood of
$(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\omega_{h+1},U_{l})=(0,0,0,0,0,G(s_{0}))$
with $\omega_{h+1}=p^{b}_{\Delta x,h+1}-p^{b}_{\Delta x,h}$ such that

$\displaystyle\delta_{1}$
$\displaystyle=\alpha_{r,1}+\alpha_{l,1}+\gamma_{1}+K_{r,1}\alpha_{r,2}+K_{\sigma,1}\Delta\sigma_{b_{h}}+K_{b,1}\omega_{h+1}+O(1)Q(\Lambda_{b})$
(4.6) $\displaystyle\delta_{2}$
$\displaystyle=\gamma_{2}+K_{r,2}\alpha_{r,2}+K_{\sigma,2}\Delta\sigma_{b_{h}}+K_{b,2}\omega_{h+1}+O(1)Q(\Lambda_{b}),$

with

$Q(\Lambda_{b})=Q^{0}((\alpha_{1},0),\gamma)+|\alpha_{1}||\Delta\sigma_{\gamma}|+|\alpha_{r,1}||\Delta\sigma_{\alpha}|,$
(4.7)

where $O(1)$ depends continuously on $M_{\infty}$. Moreover, when
$\alpha_{r,1}=\alpha_{r,2}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0$,
$p^{b}_{\Delta x,h+1}=p_{0}$, and $U_{l}=G(s_{0})$,

$\lim_{M_{\infty}\rightarrow\infty}K_{r,1}=-\dfrac{\cos^{2}(\theta_{0}+\theta_{m}^{0})}{\cos^{2}(\theta_{0}-\theta_{m}^{0})},\
\lim_{M_{\infty}\rightarrow\infty}|K_{b,i}|<\infty,\
\lim_{M_{\infty}\rightarrow\infty}K_{r,2}=0,\
\lim_{M_{\infty}\rightarrow\infty}K_{\sigma,i}=0,$ (4.8)

for $i=1,2$.

Proof. A direct computation leads to

$\displaystyle\left.\dfrac{1}{2}\frac{\partial(|\Phi(\beta_{1},0;\tilde{U}(\sigma_{b}(h);\sigma_{\alpha},U_{l}))|^{2})}{\partial\beta_{1}}\right|_{\\{\delta_{1}=\Delta\bar{\sigma}_{\alpha}=0,U_{l}=G(s_{0})\\}}=r_{1}(G(s_{0}))\cdot
G(s_{0}).$

Lemma 2.1, together with the implicit function theorem, implies that there is
a unique $\text{C}^{2}-$solution

$\beta_{1}=\beta_{1}(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\omega_{h+1};U_{l})$
(4.9)

in a neighborhood of
$(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\omega_{h+1},U_{l})=(0,0,0,0,0,G(s_{0}))$.

Using (4.1)–(4.2), we have

$\displaystyle\beta_{1}$
$\displaystyle=\beta_{1}(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\sigma_{\alpha},\omega_{h+1};U_{l})+\bar{K}_{\sigma,1}(\Delta\bar{\sigma}_{\alpha}-\Delta\sigma_{\alpha})$
$\displaystyle=\beta_{1}(\alpha_{r,1},0,\Delta\sigma_{\alpha},\Delta\sigma_{\alpha},0;U_{l})+\bar{K}_{\sigma,1}\Delta\sigma_{b_{h}}+\bar{K}_{r,1}\alpha_{r,2}+\bar{K}_{b,1}\omega_{h+1}$
$\displaystyle=\alpha_{r,1}+\bar{K}_{\sigma,1}\Delta\sigma_{b_{h}}+\bar{K}_{r,1}\alpha_{r,2}+\bar{K}_{b,1}\omega_{h+1}+O(1)|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

Taking derivative with respect to $\Delta\sigma_{b_{h}}$ in (4.5) at
$(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\omega_{h+1},U_{l})=(0,0,0,0,0,G(s_{0}))$,
we obtain

$\displaystyle G(s_{0})\cdot
r_{1}(G(s_{0}))\frac{\partial\beta_{1}}{\partial\Delta\sigma_{b_{h}}}+G(s_{0})\cdot\frac{\partial\tilde{U}(\sigma_{\alpha}+\Delta\bar{\sigma}_{\alpha}+\Delta\sigma_{b_{h}};\sigma_{\alpha},G(s_{0}))}{\partial\Delta\sigma_{b_{h}}}=0,$

which yields

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\left.\frac{\partial\beta_{1}}{\partial\Delta\sigma_{b_{h}}}\right|_{\\{\alpha_{r,1}=\alpha_{r,2}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,\,p^{b}_{\Delta
x,h+1}=p_{0},\,U_{l}=G(s_{0})\\}}=0.$

Similarly, we have

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\left.\frac{\partial\beta_{1}}{\partial\omega_{h+1}}\right|_{\\{\alpha_{r,1}=\alpha_{r,2}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,\,p^{b}_{\Delta
x,h+1}=p_{0},\,U_{l}=G(s_{0})\\}}=\lim_{M_{\infty}\rightarrow\infty}\dfrac{-p_{0}^{-\frac{1}{\gamma}}}{r_{1}(G(s_{0}))\cdot
G(s_{0})}>-\infty,$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\left.\frac{\partial\beta_{1}}{\partial\alpha_{r,2}}\right|_{\\{\alpha_{r,1}=\alpha_{r,2}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,\,p^{b}_{\Delta
x,h+1}=p_{0},\,U_{l}=G(s_{0})\\}}=-\dfrac{\cos^{2}(\theta_{0}+\theta_{m}^{0})}{\cos^{2}(\theta_{0}-\theta_{m}^{0})}.$

By the construction of the approximate solution, we have

$\displaystyle\tilde{U}(\sigma_{b}(h);\sigma_{\gamma},\Phi(\delta_{1},\delta_{2};U_{m}))=\Phi(\beta_{1},0;\tilde{U}(\sigma_{b}(h);\sigma_{\alpha},\Phi(\alpha_{l,1},0;\tilde{U}(\sigma_{\alpha};\sigma_{\gamma},\Phi(\gamma_{1},\gamma_{2};U_{m})))))$

with $U_{m}=U_{\Delta x,\vartheta}(x_{h},y_{n_{b,h-1}-2}-)$. Then, a similar
argument as in Case 4 gives (4.6)–(4.8). This completes the proof. $\square$

Lemma 4.4. In Case 4.1, for $b^{\prime}_{h}:=b_{\Delta
x,\vartheta}^{\prime}(x_{h}-)$ for $h\in\mathbb{N}_{+}$,

$\displaystyle
b^{\prime}_{h+1}-b^{\prime}_{h}=K_{c,2}\alpha_{r,2}+K_{c,\sigma}\Delta\sigma_{b_{h}}+O(1)\omega_{h+1}+O(1)|\alpha_{r,1}||\Delta\sigma_{\alpha}|$

with $O(1)$ depending continuously on $p_{0}$ such that

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}K_{c,\sigma}|_{\\{\alpha_{r,2}=\alpha_{r,1}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,p^{b}_{\Delta
x,h+1}=p_{0},U_{l}=G(s_{0})\\}}=-1,$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}K_{c,2}|_{\\{\alpha_{r,2}=\alpha_{r,1}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,p^{b}_{\Delta
x,h+1}=p_{0},U_{l}=G(s_{0})\\}}=-\frac{4}{\gamma+1}\,\frac{\cos^{2}\theta^{0}_{m}\cos^{2}(\theta^{0}_{m}+\theta_{0})}{\cos^{2}\theta_{0}}.$

Proof. From (3.2), we have

$\displaystyle b^{\prime}_{h+1}-b^{\prime}_{h}$
$\displaystyle=\frac{\Phi^{(2)}(\beta_{1}(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\omega_{h+1};U_{l}),0;\tilde{U}(\sigma_{b_{h}};\sigma_{\alpha},U_{l}))}{\Phi^{(1)}(\beta_{1}(\alpha_{r,1},\alpha_{r,2},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\omega_{h+1};U_{l}),0;\tilde{U}(\sigma_{b_{h}};\sigma_{\alpha},U_{l}))}$
$\displaystyle\quad-\frac{\tilde{U}^{(2)}(\sigma_{b_{h-1}};\sigma_{\alpha},\Phi(\alpha_{r,1},\alpha_{r,2};U_{l}))}{\tilde{U}^{(1)}(\sigma_{b_{h-1}};\sigma_{\alpha},\Phi(\alpha_{r,1},\alpha_{r,2};U_{l}))}.$

By (4.1)–(4.2), we obtain

$\displaystyle b^{\prime}_{h+1}-b^{\prime}_{h}$
$\displaystyle=\frac{\Phi^{(2)}(\beta_{1}(\alpha_{r,1},0,\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},0;U_{l}),0;\tilde{U}(\sigma_{b_{h}};\sigma_{\alpha},U_{l}))}{\Phi^{(1)}(\beta_{1}(\alpha_{r,1},0,\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},0;U_{l}),0;\tilde{U}(\sigma_{b_{h}};\sigma_{\alpha},U_{l}))}-\frac{\tilde{U}^{(2)}(\sigma_{b_{h-1}};\sigma_{\alpha},\Phi(\alpha_{r,1},0;U_{l}))}{\tilde{U}^{(1)}(\sigma_{b_{h-1}};\sigma_{\alpha},\Phi(\alpha_{r,1},0;U_{l}))}$
$\displaystyle\quad+K_{c,2}\alpha_{r,2}+O(1)\omega_{h+1}$
$\displaystyle=\frac{\Phi^{(2)}(\beta_{1}(\alpha_{r,1},0,\Delta\sigma_{\alpha},\Delta\sigma_{\alpha},0;U_{l}),0;\tilde{U}(\sigma_{b_{h-1}};\sigma_{\alpha},U_{l}))}{\Phi^{(1)}(\beta_{1}(\alpha_{r,1},0,\Delta\sigma_{\alpha},\Delta\sigma_{\alpha},0;U_{l}),0;\tilde{U}(\sigma_{b_{h-1}};\sigma_{\alpha},U_{l}))}-\frac{\tilde{U}^{(2)}(\sigma_{b_{h-1}};\sigma_{\alpha},\Phi(\alpha_{r,1},0;U_{l}))}{\tilde{U}^{(1)}(\sigma_{b_{h-1}};\sigma_{\alpha},\Phi(\alpha_{r,1},0;U_{l}))}$
$\displaystyle\quad+K_{c,\sigma}\Delta\sigma_{b_{h}}+K_{c,2}\alpha_{r,2}+O(1)\omega_{h+1}$
$\displaystyle=$ $\displaystyle
K_{c,\sigma}\Delta\sigma_{b_{h}}+K_{c,2}\alpha_{r,2}+O(1)\omega_{h+1}+O(1)|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

By similar calculation in Lemma 4.2, we have

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}K_{c,\sigma}|_{\\{\alpha_{r,2}=\alpha_{r,1}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,\,p^{b}_{\Delta
x,h+1}=p_{0},\,U_{l}=G(s_{0})\\}}=-1,$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}K_{c,2}|_{\\{\alpha_{r,2}=\alpha_{r,1}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,\,p^{b}_{\Delta
x,h+1}=p_{0},\,U_{l}=G(s_{0})\\}}=-\frac{4}{\gamma+1}\,\frac{\cos^{2}\theta^{0}_{m}\cos^{2}(\theta^{0}_{m}+\theta_{0})}{\cos^{2}\theta_{0}}.$

Then the proof is complete. $\square$

Lemma 4.5. For $\Delta x$ sufficiently small,

$\displaystyle|b^{\prime}_{h}-\sigma_{b}(h-1)|\geq 6|\Delta\sigma_{b_{h}}|,$

where $\sigma_{b}(h)=\frac{b_{\Delta x,\vartheta}(x_{h})}{x_{h}}$.

Proof. Using the notation as in Case 4.1, we have

$\displaystyle b^{\prime}_{h}=\dfrac{b_{\Delta x,\vartheta}(x_{h})-b_{\Delta
x,\vartheta}(x_{h-1})}{\Delta x}.$

Then a direct computation leads to

$\displaystyle|b^{\prime}_{h}-\sigma_{b}(h-1)|$
$\displaystyle=\left|\dfrac{b_{\Delta x,\vartheta}(x_{h})-b_{\Delta
x,\vartheta}(x_{h-1})}{\Delta x}-\sigma_{b}(h-1)\right|$
$\displaystyle=\left|\dfrac{\sigma_{b}(h)x_{h}-\sigma_{b}(h-1)x_{h-1}}{\Delta
x}-\sigma_{b}(h-1)\right|$ $\displaystyle=\left|\dfrac{x_{h}}{\Delta
x}\right||\sigma_{b}(h)-\sigma_{b}(h-1)|$ $\displaystyle\geq
6\left|\sigma_{b}(h)-\sigma_{b}(h-1)\right|$

for $\Delta x$ small enough. $\square$

Denote $\theta_{b}(h)=|\sigma_{b}(h-1)-b^{\prime}_{h}|$ that measures the
angle between boundary $\Gamma_{\Delta x,\vartheta,h}$ and the ray issuing
from the origin and passing through $(x_{h-1},b_{\Delta
x,\vartheta}(x_{h-1}))$. Then we have the following estimate for
$\theta_{b}(h)$.

Lemma 4.6. For $M_{\infty}$ sufficiently large and $\Delta x$ sufficiently
small,

$\displaystyle\theta_{b}(h)-\theta_{b}(h+1)\geq|\Delta\sigma|-|K_{c,2}||\alpha_{r,2}|-C|\omega_{h+1}|-C|\alpha_{r,1}||\Delta\sigma_{\alpha}|,$

where $h\in\mathbb{N}_{+}$, and constant $C>0$ is independent of $M_{\infty}$
and $\Delta x$.

Proof. We consider the following two different cases.

1\. $\sigma_{b}(h-1)<b^{\prime}_{h}$ so that $\sigma_{b}(h)>\sigma_{b}(h-1)$.

  * •

If $b^{\prime}_{h+1}>\sigma_{b}(h)$, then it follows from Lemma 4.2 that

$\displaystyle\theta_{b}(h)-\theta_{b}(h+1)$
$\displaystyle=b^{\prime}_{h}-\sigma_{b}(h-1)-\big{(}b^{\prime}_{h+1}-\sigma_{b}(h)\big{)}$
$\displaystyle=(1-K_{c,\sigma})\Delta\sigma_{b_{h}}-K_{c,2}\alpha_{r,2}+O(1)\omega_{h+1}+O(1)|\alpha_{r,1}||\Delta\sigma_{\alpha}|$
$\displaystyle\geq|\Delta\sigma_{b_{h}}|-|K_{c,2}||\alpha_{r,2}|-C|\omega_{h+1}|-C|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

  * •

If $b^{\prime}_{h+1}<\sigma_{b}(h)$, then, from Lemma 4.2–4.2, we have

$\displaystyle\theta_{b}(h)-\theta_{b}(h+1)$
$\displaystyle=b^{\prime}_{h}-\sigma_{b}(h-1)-\big{(}\sigma_{b}(h)-b^{\prime}_{h+1}\big{)}$
$\displaystyle=2\big{(}b^{\prime}_{h}-\sigma_{b}(h-1)\big{)}+b^{\prime}_{h+1}-\sigma_{b}(h)-\big{(}b^{\prime}_{h}-\sigma_{b}(h-1)\big{)}$
$\displaystyle\geq(11+K_{c,\sigma})|\Delta\sigma_{b_{h}}|+K_{c,2}\alpha_{r,2}+O(1)\omega_{h+1}+O(1)|\alpha_{r,1}||\Delta\sigma_{\alpha}|$
$\displaystyle\geq|\Delta\sigma_{b_{h}}|-|K_{c,2}||\alpha_{r,2}|-C|\omega_{h+1}|-C|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

2\. $\sigma_{b}(h-1)>b^{\prime}_{h}$ so that $\sigma_{b}(h)<\sigma_{b}(h-1)$.

  * •

If $b^{\prime}_{h+1}>\sigma_{b}(h)$, then it follows from Lemma 4.2–4.2 that

$\displaystyle\theta_{b}(h)-\theta_{b}(h+1)$
$\displaystyle=\sigma_{b}(h-1)-b^{\prime}_{h}-\big{(}b^{\prime}_{h+1}-\sigma_{b}(h)\big{)}$
$\displaystyle=2(\sigma_{b}(h-1)-b^{\prime}_{h})+\sigma_{b}(h)-b^{\prime}_{h+1}-(\sigma_{b}(h-1)-b^{\prime}_{h})$
$\displaystyle\geq(11+K_{c,\sigma})|\Delta\sigma_{b_{h}}|-K_{c,2}\alpha_{r,2}-O(1)\omega_{h+1}-O(1)|\alpha_{r,1}||\Delta\sigma_{\alpha}|$
$\displaystyle\geq|\Delta\sigma_{b_{h}}|-|K_{c,2}||\alpha_{r,2}|-C|\omega_{h+1}|-C|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

  * •

If $b^{\prime}_{h+1}<\sigma_{b}(h)$, then, from Lemma 4.2, we have

$\displaystyle\theta_{b}(h)-\theta_{b}(h+1)$
$\displaystyle=\sigma_{b}(h-1)-b^{\prime}_{h}-\big{(}\sigma_{b}(h)-b^{\prime}_{h+1}\big{)}$
$\displaystyle=(-1+K_{c,\sigma})\Delta\sigma_{b_{h}}+K_{c,2}\alpha_{r,2}+O(1)\omega_{h+1}+O(1)|\alpha_{r,1}||\Delta\sigma_{\alpha}|$
$\displaystyle\geq|\Delta\sigma_{b_{h}}|-|K_{c,2}||\alpha_{r,2}|-C|\omega_{h+1}|-C|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

Note that we have used the fact that

$\lim_{M_{\infty}\rightarrow\infty}K_{c,\sigma}|_{\\{\alpha_{r,2}=\beta_{1}=\Delta\sigma_{\alpha}=\Delta\bar{\sigma}_{\alpha}=\omega_{h+1}=0,p^{b}_{\Delta
x,h+1}=p_{0},U_{l}=G(s_{0})\\}}=-1$

in above estimates. This completes the proof. $\square$

Case 3. $\Lambda_{s}$ covers the part of $S_{\Delta x,\vartheta}$ but none of
$\Gamma_{\Delta x,\vartheta}$. We take three diamonds at the same time, as
shown in Fig. 4.3. Let $\Delta_{h,n_{\chi,h}-1}$, $\Delta_{h,n_{\chi,h}}$, and
$\Delta_{h,n_{\chi,h}+1}$ be the diamonds centering in
$(x_{h},y_{n_{\chi,h}-1})$, $(x_{h},y_{n_{\chi,h}})$, and
$(x_{h},y_{n_{\chi,h}+1})$, respectively. Denote
$\Lambda_{s}=\Delta_{h,n_{\chi,h}-1}\cup\Delta_{h,n_{\chi,h}}\cup\Delta_{h,n_{\chi,h}+1}$.
Let $\alpha$ and $\gamma$ be the weak waves issuing from
$(x_{h-1},y_{n_{\chi,h-1}+1})$ and $(x_{h-1},y_{n_{\chi,h-1}+2})$ respectively
and entering $\Lambda_{s}$. We divide $\alpha$ into parts
$\alpha_{l}=(\alpha_{l,1},0)$ and $\alpha_{r}=(\alpha_{r,1},\alpha_{r,2})$
where $\alpha_{l}$ and $\alpha_{r}$ enter $\Delta_{h,n_{\chi,h}}$ and
$\Delta_{h,n_{\chi,h}+1}$, respectively. Moreover, let
$\gamma=(\gamma_{1},0)$, and let $\delta$ be the outgoing wave issuing from
$(x_{h},y_{n_{\chi,h}+1})$.

Fig. 4.3. Near the strong shock wave

Then, for simplicity of notation, we denote

$\displaystyle\sigma_{\alpha}=\sigma(x_{h-1},y_{n_{\chi,h-1}+1}),$
$\displaystyle\sigma_{\chi}(h-1)=\sigma(x_{h-1},\chi_{\Delta
x,\vartheta}(x_{h-1})),$
$\displaystyle\sigma_{\chi}(h)=\sigma(x_{h},\chi_{\Delta
x,\vartheta}(x_{h})),$
$\displaystyle\Delta\sigma_{\alpha}=\sigma_{\alpha}-\sigma_{\chi}(h-1),$
$\displaystyle\Delta\bar{\sigma}_{\alpha}=\sigma_{\alpha}-\sigma_{\chi}(h),$
$\displaystyle\Delta\sigma_{\chi_{h}}=\sigma_{\chi}(h)-\sigma_{\chi}(h-1),$
$\displaystyle\sigma_{\gamma}=\sigma(x_{h-1},y_{n_{\chi,h-1}+2}),$
$\displaystyle\Delta\sigma_{\gamma}=\sigma_{\gamma}-\sigma_{\alpha}.$

To gain the estimates of $(s_{h+1},\delta)$, we need to deal with the
equation:

$\tilde{U}(\sigma_{\alpha};\sigma_{\chi}(h),\Phi(0,\beta_{2};G(s_{h+1};U_{\infty})))=\Phi(\alpha_{l,1},0;\tilde{U}(\sigma_{\alpha};\sigma_{\chi}(h-1),G(s_{h};U_{\infty}))),$
(4.10)

to obtain the following lemma.

Lemma 4.7. Equation (4.10) has a unique solution $(s_{h+1},\beta_{2})$ in a
neighborhood of

$\displaystyle(\alpha_{l,1},\alpha_{r},\gamma,\Delta\sigma_{\alpha},\Delta\sigma_{\chi_{h}},s_{h})=(0,0,0,0,0,s_{0}),$

such that

$\displaystyle\delta_{1}=\alpha_{r,1}+\gamma_{1}+\mu_{w,1}\Delta\sigma_{\chi_{h}}+K_{w,1}\alpha_{l,1}+O(1)Q(\Lambda_{s}),$
(4.11)
$\displaystyle\delta_{2}=\alpha_{r,2}+\mu_{w,2}\Delta\sigma_{\chi_{h}}+K_{w,2}\alpha_{l,1}+O(1)Q(\Lambda_{s}),$
$\displaystyle
s_{h+1}=s_{h}+K_{s}\alpha_{l,1}+\mu_{s}\Delta\sigma_{\chi_{h}},$

with

$Q(\Lambda_{s})=|\gamma_{1}||\Delta\sigma_{\gamma}|+Q^{0}(\alpha_{r},\gamma),$
(4.12)

where $O(1)$ depends continuously on $M_{\infty}$. In addition, for
$\alpha_{l}=0$, $\Delta\sigma_{\alpha}=\Delta\sigma_{\chi_{h}}=0$, and
$s_{h}=s_{0}$, denoting the derivative of $G$ by $G_{s}$, then

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}K_{w,1}=0,\quad
K_{w,2}=\dfrac{\det(r_{1}(G(s_{0})),\,G_{s}(s_{0}))}{\det(r_{2}(G(s_{0})),\,G_{s}(s_{0}))},\quad
K_{s}=\dfrac{\det(r_{2}(G(s_{0})),r_{1}(G(s_{0}))}{\det(r_{2}(G(s_{0})),G_{s}(s_{0}))},$
(4.13)
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\mu_{w,1}=0,\quad\mu_{w,2}=\dfrac{\det(\partial\tilde{U}/\partial(\Delta\sigma_{\chi_{h}}),\,G_{s}(s_{0}))}{\det(r_{2}(G(s_{0})),\,G_{s}(s_{0}))},\quad\lim_{M_{\infty}\rightarrow\infty}\mu_{s}\in(-1,0).$

Proof. From Lemma 2.1 and the implicit function theorem, (4.10) has a unique
$\text{C}^{2}$–solution $(s_{h+1},\beta_{2})$ such that

$\displaystyle
s_{h+1}=s_{h+1}(\alpha_{l,1},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\Delta\sigma_{\chi_{h}},s_{h}),$
$\displaystyle\beta_{2}=\beta_{2}(\alpha_{l,1},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\Delta\sigma_{\chi_{h}},s_{h}).$

A direct computation leads to

$\displaystyle\beta_{2}=\beta_{2}(\alpha_{l,1},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\Delta\sigma_{\chi_{h}},s_{h})$
$\displaystyle=\mu_{w,2}\Delta\sigma_{\chi_{h}}+K_{w,2}\alpha_{l,1}+\beta_{2}(0,\Delta\sigma_{\alpha},\Delta\sigma_{\alpha},0,s_{h})$
$\displaystyle=\mu_{w,2}\Delta\sigma_{\chi_{h}}+K_{w,2}\alpha_{l,1}.$

Similarly, we have

$\displaystyle
s_{h+1}=s_{h+1}(\alpha_{l,1},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\Delta\sigma_{\chi_{h}},s_{h})=\mu_{s}\Delta\sigma_{\chi_{h}}+K_{s}\alpha_{l,1}+s_{h}.$

Next, we compute the coefficients: $K_{s}$, $K_{w,2}$, $\mu_{w,2}$, and
$\mu_{s}$. Differentiating equation (4.10) with respect to $\alpha_{l,1}$ and
$\Delta\sigma_{\chi_{h}}$, and then letting
$\alpha_{l,1}=\Delta\sigma_{\alpha}=\Delta\sigma_{\chi_{h}}=0$ and
$s_{h}=s_{0}$, we can obtain

$\displaystyle r_{2}(G(s_{0}))K_{w,2}+G_{s}(s_{0})K_{s}=r_{1}(G(s_{0})),$
$\displaystyle
r_{2}(G(s_{0}))\mu_{w,2}+G_{s}(s_{0})\mu_{s}=\dfrac{\partial\tilde{U}}{\partial(\Delta\sigma_{\chi_{h}})}(\sigma_{\chi}(h);\sigma_{\chi}(h),G(s_{0})).$

Then Cramer’s rule gives the result. Moreover, since
$\theta_{0}<0<\theta^{0}_{ma}$ and
$\theta_{0}\pm\theta^{0}_{ma}\in(-\frac{\pi}{2},\frac{\pi}{2})$,

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\mu_{s}=\frac{\cos\theta_{0}\sin\theta_{m}^{0}}{\sin(\theta_{0}-\theta_{m}^{0})}\in(-1,0).$

By the construction of the approximate solution, we have

$\displaystyle\tilde{U}(\sigma_{\gamma};\sigma_{\alpha},\Phi(\delta_{1},\delta_{2};\tilde{U}(\sigma_{\alpha};\sigma_{\chi}(h),U_{m})))$
$\displaystyle=\Phi(\gamma_{1},0;\tilde{U}(\sigma_{\gamma};\sigma_{\alpha},\Phi(\alpha_{r,1},\alpha_{r,2};\tilde{U}(\sigma_{\alpha};\sigma_{\chi}(h),\Phi(0,\beta_{2}(\alpha_{l,1},\Delta\sigma_{\alpha},\Delta\bar{\sigma}_{\alpha},\Delta\sigma_{\chi_{h}},s_{h});U_{m}))))),$

with

$\displaystyle U_{m}=G(s_{h+1};U_{\infty}).$

Then, by similar arguments as in Case 4, we obtain

$\displaystyle\delta_{1}$
$\displaystyle=\alpha_{r,1}+\gamma_{1}+\mu_{w,1}\Delta\sigma_{\chi_{h}}+K_{w,1}\alpha_{l,1}+O(1)|\gamma_{1}||\Delta\sigma_{\gamma}|+O(1)Q^{0}(\alpha_{r},\gamma),$
$\displaystyle\delta_{2}$
$\displaystyle=\alpha_{r,2}+\mu_{w,2}\Delta\sigma_{\chi_{h}}+K_{w,2}\alpha_{l,1}+O(1)|\gamma_{1}||\Delta\sigma_{\gamma}|+O(1)Q^{0}(\alpha_{r},\gamma).$

This completes the proof. $\square$

Lemma 4.8. For $\Delta x$ sufficiently small,

$\displaystyle|s_{h}-\sigma_{\chi}(h-1)|\geq 6|\Delta\sigma_{\chi_{h}}|.$

Proof. Using the notation as in Case 4.2, we have

$\displaystyle\sigma_{\chi}(h)=\frac{\chi_{\Delta
x,\vartheta}(x_{h})}{x_{h}},\qquad s_{h}=\dfrac{\chi_{\Delta
x,\vartheta}(x_{h})-\chi_{\Delta x,\vartheta}(x_{h-1})}{\Delta x}.$

Then a direct computation leads to

$\displaystyle|s_{h}-\sigma_{\chi}(h-1)|$
$\displaystyle=\left|\dfrac{\chi_{\Delta x,\vartheta}(x_{h})-\chi_{\Delta
x,\vartheta}(x_{h-1})}{\Delta x}-\sigma_{\chi}(h-1)\right|$
$\displaystyle=\left|\dfrac{\sigma_{\chi}(h)x_{h}-\sigma_{\chi}(h-1)x_{h-1}}{\Delta
x}-\sigma_{\chi}(h-1)\right|$ $\displaystyle=\left|\dfrac{x_{h}}{\Delta
x}\right||\sigma_{\chi}(h)-\sigma_{\chi}(h-1)|$ $\displaystyle\geq
6\left|\sigma_{\chi}(h)-\sigma_{\chi}(h-1)\right|,$

for $\Delta x$ small enough. $\square$

Denote $\theta_{\chi}(h)=|\sigma_{\chi}(h-1)-s_{h}|$ that measures the angle
between the leading shock $S_{\Delta x,\vartheta,h}$ and the ray issuing from
the origin and passing through $(x_{h-1},\chi_{\Delta x,\vartheta}(x_{h-1}))$.
Then we have the following estimate for $\theta_{\chi}(h)$.

Lemma 4.9. For $M_{\infty}$ sufficiently large and $\Delta x$ sufficiently
small,

$\displaystyle\theta_{\chi}(h)-\theta_{\chi}(h+1)\geq|\Delta\sigma_{\chi_{h}}|-|K_{s}||\alpha_{l,1}|,$

with $h\geq 0$.

Proof. We consider the following two different cases:

1\. $\sigma_{\chi}(h-1)<s_{h}$ so that $\sigma_{\chi}(h)>\sigma_{\chi}(h-1)$.

  * •

If $s_{h+1}>\sigma_{\chi}(h)$, then it follows from Lemma 4.3 that

$\displaystyle\theta_{\chi}(h)-\theta_{\chi}(h+1)$
$\displaystyle=s_{h}-\sigma_{\chi}(h-1)-\big{(}s_{h+1}-\sigma_{\chi}(h)\big{)}=(1-\mu_{s})\Delta\sigma_{\chi_{h}}-K_{s}\alpha_{l,1}$
$\displaystyle\geq|\Delta\sigma_{\chi_{h}}|-|K_{s}||\alpha_{l,1}|.$

  * •

If $s_{h+1}<\sigma_{\chi}(h)$, then, from Lemmas 4.3–4.3, we have

$\displaystyle\theta_{\chi}(h)-\theta_{\chi}(h+1)$
$\displaystyle=s_{h}-\sigma_{\chi}(h-1)-\big{(}\sigma_{\chi}(h)-s_{h+1}\big{)}$
$\displaystyle=2(s_{h}-\sigma_{\chi}(h-1))+s_{h+1}-\sigma_{\chi}(h)-\big{(}s_{h}-\sigma_{\chi}(h-1)\big{)}$
$\displaystyle\geq(11+\mu_{s})|\Delta\sigma_{\chi_{h}}|+K_{s}\alpha_{l,1}$
$\displaystyle\geq|\Delta\sigma_{\chi_{h}}|-|K_{s}||\alpha_{l,1}|.$

2\. $\sigma_{\chi}(h-1)>s_{h}$ so that $\sigma_{\chi}(h)<\sigma_{\chi}(h-1)$.

  * •

If $s_{h+1}>\sigma_{\chi}(h)$, then it follows from Lemmas 4.3–4.3 that

$\displaystyle\theta_{\chi}(h)-\theta_{\chi}(h+1)$
$\displaystyle=\sigma_{\chi}(h-1)-s_{h}-\big{(}s_{h+1}-\sigma_{\chi}(h)\big{)}$
$\displaystyle=2(\sigma_{\chi}(h-1)-s_{h})+\sigma_{\chi}(h)-s_{h+1}-(\sigma_{\chi}(h-1)-s_{h})$
$\displaystyle\geq(11+\mu_{s})|\Delta\sigma_{\chi_{h}}|-K_{s}\alpha_{l,1}$
$\displaystyle\geq|\Delta\sigma_{\chi_{h}}|-|K_{s}||\alpha_{l,1}|.$

  * •

If $s_{h+1}<\sigma_{\chi}(h)$, then, from Lemma 4.3, we have

$\displaystyle\theta_{\chi}(h)-\theta_{\chi}(h+1)$
$\displaystyle=\sigma_{\chi}(h-1)-s_{h}-\big{(}\sigma_{\chi}(h)-s_{h+1}\big{)}$
$\displaystyle=(-1+\mu_{s})\Delta\sigma_{\chi_{h}}+K_{s}\alpha_{l,1}$
$\displaystyle\geq|\Delta\sigma_{\chi_{h}}|-|K_{s}||\alpha_{l,1}|.$

Note that we have used the fact that $\mu_{s}\in(-1,0)$ as
$M_{\infty}\rightarrow\infty$ in above estimates. This completes the proof.
$\square$

##  5\. Glimm-Type Functional and Compactness of the Approximate Solutions

For each $I\subset\cup_{k=1}^{h+1}\Omega_{\Delta x,\vartheta,k}$, there exists
$k_{I}$ with $1\leq k_{I}\leq h+1$ such that $I\cap\Gamma_{\Delta
x,\vartheta,k_{I}}\neq\emptyset$. Next, as in [51, 38], we assign each mesh
curve $I\subset\bigcup_{k=1}^{h+1}\Omega_{\Delta x,\vartheta,k}$ with a Glimm-
type functional $F_{s}(I)$; see also [48, 10]:

Definition 5.1. (Weighted total variation). Define

$\displaystyle L_{0}^{(i)}(I)=\sum\\{|\alpha_{i}|:\alpha_{i}\text{ is the weak
$i$-wave crossing $I$}\\}\qquad\mbox{for $i=1,\,2$},$ $\displaystyle
L_{1}(I)=\sum\\{|\omega_{k}|:k>k_{I}\\},$ $\displaystyle
L_{s}(I)=\theta_{\chi}(I)\quad\text{ for $\theta_{\chi}(I)=\theta_{\chi}(h)$
in Lemma \ref{Lem:Estchashock} when $S_{\Delta x,\vartheta}$ crossing $I$},$
$\displaystyle L_{b}(I)=\theta_{b}(I)\quad\,\text{ for
$\theta_{b}(I)=\theta_{b}(h)$ in Lemma \ref{Lem:Estchabdry} when
$\Gamma_{\Delta x,\vartheta}$ crossing $I$}.$

Then the weighted total variation is defined as

$\displaystyle
L(J)=L_{0}^{(1)}(I)+K_{2}L_{0}^{(2)}(I)+K_{1}L_{1}(I)+K_{3}L_{s}(I)+K_{4}L_{b}(I),$

where $K_{l}$ are positive constants for $l=1,2,3,4$.

Let

$\sigma^{*}=b_{0}+C_{1}\sum_{h=1}^{\infty}|\omega_{h}|,\qquad\sigma_{*}=s_{0}-\varpi,$
(5.1)

where $s_{0}$ is the velocity of the leading shock of the background solution,
$\varpi$ and $C_{1}$ are constants to be determined; see also [18, 48, 10].
Note that $\varpi$ and $\sum_{h\geq 1}|\omega_{h}|$ are chosen so small that
the largeness of $M_{\infty}$ implies the smallness of $b_{0}-s_{0}$, which
leads to the smallness of $\sigma^{*}-\sigma_{*}$. We now define the total
interaction potential.

Definition 5.2. (Total interaction potential). Define

$\displaystyle Q_{0}(I)=\sum\\{|\alpha||\beta|\,:\,\text{$\alpha$ and $\beta$
are weak waves crossing $I$ and approach}\\},$ $\displaystyle
Q_{1}(I)=\sum\\{|\alpha||\sigma_{\alpha}-\sigma_{*}|\,:\,\text{$\alpha$ is a
weak $1-$wave crossing $I$}\\},$ $\displaystyle
Q_{2}(I)=\sum\\{|\alpha||\sigma^{*}-\sigma_{\alpha}|\,:\,\text{$\alpha$ is a
weak $2-$wave crossing $I$}\\},$

where $\sigma_{\alpha}$ is the $\sigma$-coordinate of the grid point where
$\alpha$ issues. Then the total interaction potential is defined as

$\displaystyle Q(I)=Q_{0}(I)+2Q_{1}(I)+2Q_{2}(I).$

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

Definition 5.3. (Glimm-type functional). Let

$\displaystyle F(I)=L(I)+KQ(I),$

where $K$ is a big real number to be chosen later.

Let

$E_{\Delta x,\vartheta}(\Lambda)=\left\\{\begin{aligned}
&Q(\Lambda)&&\text{(defined in Case \ref{case:1}),}&\\\
&\xi\big{(}|\alpha_{r,2}|+|\omega_{h+1}|+|\Delta\sigma_{b_{h}}|+Q(\Lambda_{b})\big{)}&&\text{(defined
in Case \ref{case:2}),}&\\\
&\xi\big{(}|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|+Q(\Lambda_{s})\big{)}&&\text{(defined
in Case \ref{case:3}),}&\end{aligned}\right.$ (5.2)

with $\xi>0$ sufficiently small and to be chosen later.

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

Lemma 5.1. Let $K_{r,1}$, $K_{w,2}$, $K_{s}$ and $\mu_{w,2}$ be given by
Lemmas 4.2 and 4.3. Then

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\big{(}|K_{r,1}||K_{w,2}|+|K_{r,1}||K_{s}||\mu_{w,2}|\big{)}<1.$

Proof. Lemmas 2.1–2.1 give

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}|K_{r,1}||K_{w,2}|=\left|\dfrac{\sin(\theta_{0}+\theta_{m}^{0})}{\sin(\theta_{0}-\theta_{m}^{0})}\right|,$
$\displaystyle\lim_{M_{\infty}\rightarrow\infty}|K_{r,1}||K_{s}||\mu_{w,2}|$
$\displaystyle\,\,\,=\dfrac{\cos^{2}(\theta_{0}+\theta_{m}^{0})}{\cos^{2}(\theta_{0}-\theta_{m}^{0})}\lim_{M_{\infty}\rightarrow\infty}\left|\dfrac{\det\big{(}r_{1}(U),r_{2}(U)\big{)}}{\det\big{(}r_{2}(G(s_{0}),G^{\prime}(s_{0}))\big{)}}\right|\left|\dfrac{\det((\partial\tilde{U})/(\partial\Delta\sigma_{\chi_{h}}),G_{s}(s_{0};U_{\infty}))}{\det(r_{2}(G(s_{0};U_{\infty})),G_{s}(s_{0};U_{\infty}))}\right|$
$\displaystyle\,\,\,=\dfrac{\sin
2\theta_{m}^{0}\cos\theta_{0}|\sin\theta_{0}|}{\sin^{2}(\theta_{0}-\theta_{m}^{0})}.$

Note that $\theta_{0}\in(-\frac{\pi}{2},0)$ and
$\theta_{0}\pm\theta_{m}^{0}\in(-\frac{\pi}{2},\frac{\pi}{2})$, and that
$\theta_{m}^{0}\in(0,\frac{\pi}{2})$. Then, when
$\theta_{0}+\theta_{m}^{0}<0$,

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\big{(}|K_{r,1}||K_{w,2}|+|K_{r,1}||K_{s}||\mu_{w,2}|\big{)}<\dfrac{2\sin\theta_{m}^{0}\cos\theta_{0}-\sin(\theta_{0}+\theta_{m}^{0})}{\sin(\theta_{m}^{0}-\theta_{0})}=1;$

when $\theta_{0}+\theta_{m}^{0}>0$,

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\big{(}|K_{r,1}||K_{w,2}|+|K_{r,1}||K_{s}||\mu_{w,2}|\big{)}<\dfrac{2\cos\theta_{m}^{0}|\sin\theta_{0}|+\sin(\theta_{0}+\theta_{m}^{0})}{\sin(\theta_{m}^{0}-\theta_{0})}=1.$

This implies the expected result. $\square$

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

Lemma 5.2. There exist positive constants $K_{2}$ and $K_{3}$ such that

$\displaystyle\lim_{M_{\infty}\rightarrow\infty}\big{(}K_{2}|K_{w,2}|+K_{3}|K_{s}|\big{)}<1,\quad\lim_{M_{\infty}\rightarrow\infty}\big{(}K_{2}|\mu_{w,2}|-K_{3}\big{)}<0,\quad\lim_{M_{\infty}\rightarrow\infty}\big{(}K_{2}-|K_{r,1}|\big{)}>0.$

Proof. Let $K_{r,1}^{*}=\lim_{M_{\infty}\rightarrow\infty}|K_{r,1}|$,
$K_{w,2}^{*}=\lim_{M_{\infty}\rightarrow\infty}|K_{w,2}|$,
$K_{s}^{*}=\lim_{M_{\infty}\rightarrow\infty}|K_{s}|$, and
$\mu_{w,2}^{*}=\lim_{M_{\infty}\rightarrow\infty}|\mu_{w,2}|$. Then, by Lemma
5,

$\displaystyle K_{r,1}^{*}\big{(}K_{w,2}^{*}+K_{s}^{*}\mu_{w,2}^{*}\big{)}<1.$

Hence, we choose $K_{2}$ such that

$\displaystyle K_{2}>K_{r,1}^{*},\qquad
K_{2}\big{(}K_{w,2}^{*}+K_{s}^{*}\mu_{w,2}^{*}\big{)}<1,$

which implies

$\displaystyle K_{2}K_{s}^{*}\mu_{w,2}^{*}<1-K_{2}K_{w,2}^{*}.$

Then we take $K_{3}$ such that

$\displaystyle K_{3}>K_{2}\mu_{w,2}^{*},\qquad
K_{3}K_{s}^{*}<1-K_{2}K_{w,2}^{*},$

and the proof is complete. $\square$

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

Proposition 5.1. Let $M_{\infty}$ be sufficiently large, and let
$\sigma^{*}-\sigma_{*}$ and $\sum_{h\geq 1}|\omega_{h}|$ be sufficiently
small. Let $I$ and $J$ be a pair of space-like mesh curves with $J$ being an
immediate successor of $I$. The region bounded by the difference between $I$
and $J$ is denoted as $\Lambda$. Then there exist positive constants
$\epsilon_{\infty}$, $K$, and $K_{l}$ for $l=1,2,3,4$, such that, if
$F(I)<\epsilon_{\infty}$, then

$\displaystyle F(I)\leq F(J)-\frac{1}{4}E_{\Delta x,\vartheta}(\Lambda),$

where $E_{\Delta x,\vartheta}(\Lambda)$ is given by (5.2).

Proof. When $M_{\infty}$ is large enough, according to Lemma 5, there are
constants $K_{2}$ and $K_{3}$ so that

$\displaystyle K_{2}|K_{w,2}|+K_{3}|K_{s}|<1-\xi_{0},\quad
K_{2}|\mu_{w,2}|-K_{3}<-\xi_{0},\quad K_{2}-|K_{r}|-K_{4}|K_{c,2}|>\xi_{0},$

for some $\xi_{0}>0$.

Now, as in [38], we prove the result inductively; see also [48, 10]. We
consider three special cases as in §4, depending on the location of $\Lambda$.
From now on, we use $C$ to denote a universal constant depending only on the
system, which may be different at each occurrence.

Case 1. $\Lambda$ lies between $\Gamma_{\Delta x,\vartheta}$ and $S_{\Delta
x,\vartheta}$. We consider the case as in Lemma 4.1. Notice that

$\displaystyle(L_{0}^{(1)}+K_{2}L_{0}^{(2)})(J)-(L_{0}^{(1)}+K_{2}L_{0}^{(2)})(I)\leq
CQ(\Lambda),$ $\displaystyle L_{b}(J)-L_{b}(I)=0,$
$\displaystyle(K_{1}L_{1}+K_{3}L_{s})(J)-(K_{1}L_{1}+K_{3}L_{s})(I)=0.$

Then we obtain

$\displaystyle L(J)-L(I)\leq CQ(\Lambda).$

For the terms contained in $Q$, we have

$\displaystyle Q_{0}(J)-Q_{0}(I)\leq CL(I)Q(\Lambda)-Q^{0}(\Lambda).$

For Case 1.1:

$\displaystyle(Q_{1}+Q_{2})(J)-(Q_{1}+Q_{2})(I)$
$\displaystyle=|\delta_{1}|(\sigma_{2}-\sigma_{*})-|\alpha_{1}|(\sigma_{2}-\sigma_{*})-|\beta_{1}|(\sigma_{1}-\sigma_{*})$
$\displaystyle\quad+|\delta_{2}|(\sigma^{*}-\sigma_{2})-|\alpha_{2}|(\sigma^{*}-\sigma_{2})$
$\displaystyle\leq
C(\sigma^{*}-\sigma_{*})Q(\Lambda)-|\Delta\sigma||\beta_{1}|.$

For Case 1.2:

$\displaystyle(Q_{1}+Q_{2})(J)-(Q_{1}+Q_{2})(I)$
$\displaystyle\leq|\delta_{1}|(\sigma_{1}-\sigma_{*})-|\beta_{1}|(\sigma_{1}-\sigma_{*})$
$\displaystyle\quad+|\delta_{2}|(\sigma^{*}-\sigma_{1})-|\alpha_{2}|(\sigma^{*}-\sigma_{2})-|\beta_{2}|(\sigma^{*}-\sigma_{1})$
$\displaystyle\leq
C(\sigma^{*}-\sigma_{*})Q(\Lambda)-|\Delta\sigma||\alpha_{2}|,$

which gives

$\displaystyle Q(J)-Q(I)\leq$
$\displaystyle-\big{(}1-C(L(I)+\sigma^{*}-\sigma_{*})\big{)}Q(\Lambda).$

When $L(I)$ and $\sigma^{*}-\sigma_{*}$ are small enough, and $K$ is
sufficiently large, it follows that

$\displaystyle
F(J)-F(I)\leq-\left\\{K\big{(}1-C(L(I)+\sigma^{*}-\sigma_{*})\big{)}-C\right\\}Q(\Lambda)\leq-\frac{1}{4}Q(\Lambda).$

Case 2.
$\Lambda_{b}=\Delta_{h,n_{b,h}-1}\cup\Delta_{h,n_{b,h}}\cup\Delta_{h,n_{b,h}+1}$
covers a part of $\Gamma_{\Delta x,\vartheta}$ but none of $S_{\Delta
x,\vartheta}$. Direct computation shows that

$\displaystyle
L_{0}^{(1)}(J)-L_{0}^{(1)}(I)\leq|K_{r,1}||\alpha_{r,2}|+|K_{\sigma,1}||\Delta\sigma_{b_{h}}|+|K_{b,1}||\omega_{h+1}|+CQ(\Lambda_{b}),$
$\displaystyle
L_{0}^{(2)}(J)-L_{0}^{(2)}(I)\leq-|\alpha_{r,2}|+|K_{r,2}||\alpha_{r,2}|+|K_{\sigma,2}||\Delta\sigma_{b_{h}}|+|K_{b,2}||\omega_{h+1}|+CQ(\Lambda_{b}),$
$\displaystyle L_{1}(J)-L_{1}(I)=-|\omega_{h+1}|,$ $\displaystyle
L_{s}(J)-L_{s}(I)=0,$ $\displaystyle
L_{b}(J)-L_{b}(I)=-|\Delta\sigma_{b_{h}}|+|K_{c,2}||\alpha_{r,2}|+C|\omega_{h+1}|+C|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

Combining the above estimates together, we obtain

$\displaystyle L(J)-L(I)$
$\displaystyle\leq-\big{(}K_{2}-|K_{r,1}|-K_{4}|K_{c,2}|-K_{2}|K_{r,2}|\big{)}|\alpha_{r,2}|$
$\displaystyle\quad-\big{(}K_{1}-|K_{b,1}|-K_{2}|K_{b,2}|-CK_{4}\big{)}|\omega_{h+1}|$
$\displaystyle\quad-\big{(}K_{4}-|K_{\sigma,1}|-K_{2}|K_{\sigma,2}|\big{)}|\Delta\sigma_{b_{h}}|+CQ(\Lambda_{b})+C|\alpha_{r,1}||\Delta\sigma_{\alpha}|.$

For the terms contained in $Q$, noting that
$|\Delta\sigma_{\alpha}|\leq|\Delta\sigma_{\gamma}|$, we have

$\displaystyle Q_{0}(J)-Q_{0}(I)$
$\displaystyle\leq-Q^{0}((\alpha_{1},0),\gamma)+CL(I)\big{(}|\alpha_{r,2}|+|\Delta\sigma_{b_{h}}|+|\omega_{h+1}|+CQ(\Lambda_{b})\big{)},$
$\displaystyle Q_{1}(J)-Q_{1}(I)$
$\displaystyle=|\delta_{1}|(\sigma_{\gamma}-\sigma_{*})-(|\alpha_{l,1}|+|\alpha_{r,1}|)(\sigma_{\alpha}-\sigma_{*})-|\gamma_{1}|(\sigma_{\gamma}-\sigma_{*})$
$\displaystyle\leq-(|\alpha_{l,1}|+|\alpha_{r,1}|)|\Delta\sigma_{\gamma}|+C(\sigma^{*}-\sigma_{*})\big{(}|\alpha_{r,2}|+|\Delta\sigma_{b_{h}}|+|\omega_{h+1}|+CQ(\Lambda_{b})\big{)},$
$\displaystyle Q_{2}(J)-Q_{2}(I)$
$\displaystyle=|\delta_{2}|(\sigma^{*}-\sigma_{\gamma})-|\alpha_{r,2}|(\sigma^{*}-\sigma_{\alpha})-|\gamma_{2}|(\sigma^{*}-\sigma_{\gamma})$
$\displaystyle\leq
C(\sigma^{*}-\sigma_{*})\big{(}|\alpha_{r,2}|+|\Delta\sigma_{b_{h}}|+|\omega_{h+1}|+CQ(\Lambda_{b})\big{)}.$

Then we conclude

$\displaystyle Q(J)-Q(I)$
$\displaystyle\leq-Q^{0}((\alpha_{1},0),\gamma)+CL(I)\big{(}|\alpha_{r,2}|+|\Delta\sigma_{b_{h}}|+|\omega_{h+1}|+CQ(\Lambda_{b})\big{)}-|\alpha_{1}||\Delta\sigma_{\gamma}|$
$\displaystyle\quad-|\alpha_{r,1}||\Delta\sigma_{\alpha}|+2C(\sigma^{*}-\sigma_{*})\big{(}|\alpha_{r,2}|+|\Delta\sigma_{b_{h}}|+|\omega_{h+1}|+CQ(\Lambda_{b})\big{)}$
$\displaystyle\leq-\big{(}1-C(L(I)+\sigma^{*}-\sigma_{*})\big{)}Q(\Lambda_{b})+C\big{(}L(I)+\sigma^{*}-\sigma_{*}\big{)}\big{(}|\alpha_{r,2}|+|\Delta\sigma_{b_{h}}|+|\omega_{h+1}|\big{)}.$

Finally, combining all the estimates above together, we obtain

$\displaystyle F(J)-F(I)$
$\displaystyle\leq-\left\\{K\big{(}1-C(L(I)+\sigma^{*}-\sigma_{*})\big{)}-C\right\\}Q(\Lambda_{b})$
$\displaystyle\quad-\left\\{K_{2}-|K_{r,1}|-K_{4}|K_{c,2}|-K_{2}|K_{r,2}|-KC\big{(}L(I)+\sigma^{*}-\sigma_{*}\big{)}\right\\}|\alpha_{r,2}|$
$\displaystyle\quad-\left\\{K_{1}-|K_{b,1}|-K_{2}|K_{b,2}|-CK_{4}-KC\big{(}L(I)+\sigma^{*}-\sigma_{*}\big{)}\right\\}|\omega_{h+1}|$
$\displaystyle\quad-\left\\{K_{4}-|K_{\sigma,1}|-K_{2}|K_{\sigma,2}|-KC\big{(}L(I)+\sigma^{*}-\sigma_{*}\big{)}\right\\}|\Delta\sigma_{b_{h}}|.$

Taking suitably large $K_{1}$, then, when $K$ is sufficiently large, and
$L(I)$ and $\sigma^{*}-\sigma_{*}$ are sufficiently small, we conclude

$\displaystyle
F(J)-F(I)\leq-\frac{\xi}{4}\big{(}|\alpha_{r,2}|+|\omega_{h+1}|+|\Delta\sigma_{b_{h}}|+Q(\Lambda_{b})\big{)},$

for some $\xi>0$ small enough.

Case 3.
$\Lambda_{s}=\Delta_{h,n_{\chi,h}-1}\cup\Delta_{h,n_{\chi,h}}\cup\Delta_{h,n_{\chi,h}+1}$
covers a part of $S_{\Delta x,\vartheta}$ but none of $\Gamma_{\Delta
x,\vartheta}$. A direct computation shows that

$\displaystyle
L_{0}^{(1)}(J)-L_{0}^{(1)}(I)\leq-|\alpha_{l,1}|+|K_{w,1}||\alpha_{l,1}|+|\mu_{w,1}||\Delta\sigma_{\chi_{h}}|+CQ(\Lambda_{s}),$
$\displaystyle
L_{0}^{(2)}(J)-L_{0}^{(2)}(I)\leq|K_{w,2}||\alpha_{l,1}|+|\mu_{w,2}||\Delta\sigma_{\chi_{h}}|+CQ(\Lambda_{s}),$
$\displaystyle L_{1}(J)-L_{1}(I)=0,$ $\displaystyle
L_{s}(J)-L_{s}(I)\leq-|\Delta\sigma_{\chi_{h}}|+|K_{s}||\alpha_{l,1}|,$
$\displaystyle L_{b}(J)-L_{b}(I)=0.$

Combine the above estimates together, we obtain

$\displaystyle L(J)-L(I)$
$\displaystyle\leq-(1-|K_{w,1}|-K_{2}|K_{w,2}|-K_{3}|K_{s}|)|\alpha_{l,1}|-\big{(}K_{3}-|\mu_{w,1}|-K_{2}|\mu_{w,2}|\big{)}|\Delta\sigma_{\chi_{h}}|+CQ(\Lambda_{s})$
$\displaystyle\leq-\big{(}1-K_{2}|K_{w,2}|-K_{3}|K_{s}|-|K_{w,1}|\big{)}|\alpha_{l,1}|-\big{(}K_{3}-K_{2}|\mu_{w,2}|-|\mu_{w,1}|\big{)}|\Delta\sigma_{\chi_{h}}|+CQ(\Lambda_{s}).$

For the terms contained in $Q$, we have

$\displaystyle Q_{0}(J)-Q_{0}(I)$
$\displaystyle\leq-Q^{0}(\alpha_{r},\gamma)+CL(I)\big{(}|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|+Q(\Lambda_{s})\big{)},$
$\displaystyle Q_{1}(J)-Q_{1}(I)$
$\displaystyle=|\delta_{1}|(\sigma_{\alpha}-\sigma_{*})-(|\alpha_{l,1}|+|\alpha_{r,1}|)(\sigma_{\alpha}-\sigma_{*})-|\gamma_{1}|(\sigma_{\gamma}-\sigma_{*})$
$\displaystyle\leq-|\gamma_{1}||\Delta\sigma_{\gamma}|+C(\sigma^{*}-\sigma_{*})\big{(}|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|+Q(\Lambda_{s})\big{)},$
$\displaystyle Q_{2}(J)-Q_{2}(I)$
$\displaystyle=|\delta_{2}|(\sigma^{*}-\sigma_{\gamma})-|\alpha_{r,2}|(\sigma^{*}-\sigma_{\alpha})$
$\displaystyle\leq
C(\sigma^{*}-\sigma_{*})\big{(}|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|+Q(\Lambda_{s})\big{)}.$

Then we deduce that

$\displaystyle
Q(J)-Q(I)\leq-\big{(}1-C(L(I)+\sigma^{*}-\sigma_{*})\big{)}Q(\Lambda_{s})+C\big{(}L(I)+\sigma^{*}-\sigma_{*}\big{)}\big{(}|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|\big{)}.$

Finally, combining all the estimates above together, we obtain

$\displaystyle F(J)-F(I)\leq$
$\displaystyle-\left\\{K\big{(}1-C(L(I)+\sigma^{*}-\sigma_{*})\big{)}-C\right\\}Q(\Lambda_{s})$
$\displaystyle-\Big{\\{}1-K_{2}|K_{w,2}|-K_{3}|K_{s}|-|K_{w,1}|-CK\big{(}L(I)+\sigma^{*}-\sigma_{*}\big{)}\Big{\\}}|\alpha_{l,1}|$
$\displaystyle-\Big{\\{}K_{3}-K_{2}|\mu_{w,2}|-|\mu_{w,1}|-CK\big{(}L(I)+\sigma^{*}-\sigma_{*}\big{)}\Big{\\}}|\Delta\sigma_{\chi_{h}}|.$

When $K$ is sufficiently large, and $L(I)$ and $\sigma^{*}-\sigma_{*}$ are
sufficiently small, we conclude that

$\displaystyle F(J)-F(I)\leq$
$\displaystyle-\frac{\xi}{4}(Q(\Lambda_{s})+|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|)$

for some $\xi>0$ small enough. Combining the above three cases, we conclude
our result. $\square$

Now, let $I_{h}$ be the mesh curve in the stripe: $\\{(x,y)\,:\,x_{h-1}\leq
x\leq x_{h}\\}$ for $h\in\mathbb{N}_{+}$; that is, $I_{h}$ connects all the
mesh points in the strip. Let $I$ and $J$ be any pair of mesh curves with
$I_{h}<I<J<I_{h+1}$, and let $J$ be an immediate successor of $I$. That is,
the mesh points on $J$ differ from those on $I$ by only one point generally
(except three points near the approximate boundary or near the approximate
shock), and the region bounded by the difference between $I$ and $J$ is
denoted by $\Lambda$. Proposition 5 suggests that the total variation of the
approximate solution is uniformly bounded.

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

Proposition 5.2. There exists a constant $\bar{C}>0$, independent of $\Delta
x$, $\vartheta$, and $U_{\Delta x,\vartheta}$, such that

$\displaystyle\text{T.V.}\\{s_{\Delta
x,\vartheta}:[0,\infty)\\}=\sum_{h=0}^{\infty}|s_{h+1}-s_{h}|\leq\bar{C}\sum_{h\geq
1}|\omega_{h}|,$ $\displaystyle\text{T.V.}\\{b^{\prime}_{\Delta
x,\vartheta}:[0,\infty)\\}=\sum_{h=0}^{\infty}|b^{\prime}_{h+1}-b^{\prime}_{h}|\leq\bar{C}\sum_{h\geq
1}|\omega_{h}|.$

Proof. Notice that

$\displaystyle\text{T.V.}\\{s_{\Delta x,\vartheta}:[0,\infty)\\}$
$\displaystyle=\sum_{h=0}^{\infty}|s_{h+1}-s_{h}|\leq
O(1)\sum_{\Lambda_{s}}E_{\Delta x,\vartheta}(\Lambda_{s})$ $\displaystyle\leq
O(1)\sum_{\Lambda}F(I)-F(J)\leq O(1)F(I_{1}).$

Similarly, we have

$\displaystyle\text{T.V.}\\{b^{\prime}_{\Delta x,\vartheta}:[0,\infty)\\}\leq
O(1)F(I_{1}).$

Therefore, $\bar{C}$ in the statement can be determined. $\square$

We choose $C_{1}=2\bar{C}$ and $\varpi=2\bar{C}\sum_{h\geq 1}|\omega_{h}|$ in
(5.1). The largeness of $M_{\infty}$ and the smallness of $\sum_{h\geq
1}|\omega_{h}|$ imply the smallness of $\sigma^{*}-\sigma_{*}$. Then,
following [14, 52], we conclude

Theorem 5.1. Under assumptions (1.2)–(1.2), if $M_{\infty}$ is sufficiently
large and $\sum_{h\geq 1}|\omega_{h}|$ is sufficiently small, then, for any
$\vartheta\in\Pi_{h=0}^{\infty}[0,1)$ and $\Delta x>0$, the modified Glimm
scheme introduced above defines a sequence of global approximate solutions
$U_{\Delta x,\vartheta}(x,y)$ such that

$\displaystyle\sup_{x>0}\text{T.V.}\\{U_{\Delta
x,\vartheta}(x,y):(-\infty,b_{\Delta x,\vartheta}(x))\\}<\infty,$
$\displaystyle\int_{-\infty}^{0}|U_{\Delta x,\vartheta}(x_{1},y+b_{\Delta
x,\vartheta}(x_{1}))-U_{\Delta x,\vartheta}(x_{2},y+b_{\Delta
x,\vartheta}(x_{2}))|\,{\rm d}y<L_{1}|x_{1}-x_{2}|,$

for some $L_{1}>0$ independent of $U_{\Delta x,\vartheta}$, $\Delta x$, and
$\vartheta$.

##  6\. Convergence of the Approximate Solutions

In §5, the uniform bound of the total variation of the approximate solutions
$U_{\Delta x,\vartheta}$ has been obtained. Then, by Propositions 5–5, the
existence of convergent subsequences of the approximate solutions
$\\{U_{\Delta x,\vartheta}\\}$ follows. Now we are going to prove that there
is a convergent subsequence of the approximate solutions $\\{U_{\Delta
x,\vartheta}\\}$ whose limit is an entropy solution to our problem.

Take $\Delta x=2^{-m}$, $m=0,1,2,\cdots$. For any randomly chosen sequences
$\vartheta=(\vartheta_{0},\vartheta_{1},\vartheta_{2},\dots,\vartheta_{h},\dots)$,
we obtain a set of approximate solutions, which are denoted by
$\\{(u_{m},v_{m})\\}$. It suffices to prove that there is a subsequence (still
denoted by) $\\{(u_{m},v_{m})\\}$ such that

$\iint_{\Omega_{\Delta
x,\vartheta}}\Big{(}\phi_{x}\rho_{m}u_{m}+\phi_{y}\rho_{m}v_{m}-\frac{\rho_{m}v_{m}\phi}{y}\Big{)}\,{\rm
d}x{\rm
d}y+\int_{-\infty}^{y_{0}(0)}\phi(x_{0},y)\rho(x_{0},y)u(x_{0},y)\,{\rm
d}y\rightarrow 0$ (6.1)

for any $\phi(x,y)\in\text{C}_{0}^{1}(\mathbb{R}^{2};\mathbb{R})$, and

$\iint_{\Omega_{\Delta
x,\vartheta}}\left(\phi_{x}v_{m}-\phi_{y}u_{m}\right){\rm d}x{\rm
d}y\rightarrow 0$ (6.2)

for any $\phi(x,y)\in\text{C}_{0}^{1}(\Omega;\mathbb{R})$, as
$m\rightarrow\infty$. We now prove (6.1) only, since (6.2) can be deduced
analogously.

For simplicity, we drop the subscript of $(u_{m},v_{m})$, and rewrite (6.1) as

$\displaystyle\iint_{\Omega_{\Delta x,\vartheta}}\Big{(}\phi_{x}\rho
u+\phi_{y}\rho v-\frac{\rho v\phi}{y}\Big{)}\,{\rm d}x{\rm
d}y+\int_{-\infty}^{y_{0}(0)}\phi(x_{0},y)\rho(x_{0},y)u(x_{0},y)\,{\rm d}y$
$\displaystyle=\sum_{h=1}^{\infty}\iint_{\Omega_{\Delta
x,\vartheta,h}}\Big{(}\phi_{x}\rho u+\phi_{y}\rho v-\frac{\rho
v\phi}{y}\Big{)}\,{\rm d}x{\rm
d}y+\int_{-\infty}^{y_{0}(0)}\phi(x_{0},y)\rho(x_{0},y)u(x_{0},y)\,{\rm d}y.$

By the shock waves and the upper/lower edges of rarefaction waves, each
$\Omega_{\Delta x,\vartheta,h}$ can be divided into smaller polygons:
$\Omega_{\Delta x,\vartheta,h,j}$, $j=0,-1,-2,\cdots$, alternatively, where
$\Omega_{\Delta x,\vartheta,h,0}$ is the uppermost area below the approximate
boundary $\Gamma_{\Delta x,\vartheta,h}$. Then we have

$\displaystyle\iint_{\Omega_{\Delta x,\vartheta}}\Big{(}\phi_{x}\rho
u+\phi_{y}\rho v-\frac{\rho v\phi}{y}\Big{)}\,{\rm d}x{\rm
d}y+\int_{-\infty}^{y_{0}(0)}\phi(x_{0},y)\rho(x_{0},y)u(x_{0},y)\,{\rm d}y$
(6.3)
$\displaystyle=\sum_{h=1}^{\infty}\sum_{j=0}^{-\infty}\iint_{\Omega_{\Delta
x,\vartheta,h,j}}\Big{(}\phi_{x}\rho u+\phi_{y}\rho v-\frac{\rho
v\phi}{y}\Big{)}\,{\rm d}x{\rm
d}y+\int_{-\infty}^{y_{0}(0)}\phi(x_{0},y)\rho(x_{0},y)u(x_{0},y)\,{\rm d}y$
$\displaystyle=-\sum_{h,j}\iint_{\Omega_{\Delta
x,\vartheta,h,j}}\phi\Big{(}(\rho u)_{x}+(\rho v)_{y}+\frac{\rho
v}{y}\Big{)}\,{\rm d}x{\rm d}y+\sum_{h,j}\iint_{\Omega_{\Delta
x,\vartheta,h,j}}\big{(}(\phi\rho u)_{x}+(\phi\rho v)_{y}\big{)}\,{\rm d}x{\rm
d}y$
$\displaystyle\quad+\int_{-\infty}^{y_{0}(0)}\phi(x_{0},y)\rho(x_{0},y)u(x_{0},y)\,{\rm
d}y$ $\displaystyle=:\text{I}+\text{II}+\text{III}.$

We first have

Proposition 6.1. I$\rightarrow 0$ as $\Delta x\rightarrow 0$.

Proof. To deal with the first term I in (6.3), we use the transform:

$\sigma=\frac{y}{x},\qquad\eta=\frac{y-y_{n}(h)}{x-x_{h}},$ (6.4)

where $(x_{h},y_{n}(h))$ are the center of the Riemann problem, and $n$
depends on $j$. Then we obtain

$\displaystyle\text{I}=$ $\displaystyle-\sum_{h,j}\iint_{\Omega_{\Delta
x,\vartheta,h,j}}\frac{(\sigma-\eta)\phi\rho}{\sigma(y_{n}(h)-\eta
x_{h})}\Big{(}-\sigma^{2}\big{(}1-\frac{u^{2}}{c^{2}}\big{)}u_{\sigma}+\frac{2uv\sigma^{2}}{c^{2}}v_{\sigma}+\big{(}1-\frac{v^{2}}{c^{2}}\big{)}v_{\sigma}\sigma+v\Big{)}\,{\rm
d}\eta{\rm d}\sigma$ (6.5) $\displaystyle-\sum_{h,j}\iint_{\Omega_{\Delta
x,\vartheta,h,j}}\frac{(\eta-\sigma)\phi}{\sigma
x_{h}-y_{n}(h)}\big{(}-\eta(\rho u)_{\eta}+(\rho v)_{\eta}\big{)}\,{\rm
d}\eta{\rm d}\sigma.$

From the construction of the approximate solutions, the first term of (6.5)
vanishes. For the second term, we have

$\displaystyle-\eta(\rho u)_{\eta}+(\rho v)_{\eta}=O(1)\Delta\sigma,$

where $\Delta\sigma$ is the change of the $\sigma-$coordinate in domain
$\Omega_{\Delta x,\vartheta,h,j}$. Denote the rarefaction waves in
$\Omega_{\Delta x,\vartheta,h}$ alternatively by $\alpha_{R,h,i}$. Then we
have

$\displaystyle\text{I}=O(1)\sum_{h,j}\Delta\eta(\Delta\sigma)^{2},$

with $\Delta\eta=O(1)\alpha_{R,h,i}$. According to Proposition 5, the total
strength $\sum_{i}|\alpha_{R,h,i}|$ of rarefaction waves in $\Omega_{\Delta
x,\vartheta,h}$ is bounded, so that

$\text{I}=O(1)\,\text{diam}(\text{supp}\phi)\,\Delta x,$ (6.6)

which gives desired result. $\square$

Next, applying Green’s formula in each $\Omega_{\Delta x,\vartheta,h,j}$, we
obtain

$\displaystyle\text{II}+\text{III}$
$\displaystyle=\sum_{h=1}^{\infty}\int_{-\infty}^{b_{\Delta
x,\vartheta}(x_{h})}\phi(x_{h},y)\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ (6.7)
$\displaystyle\quad+\sum_{h=0}^{\infty}\int_{x_{h}}^{x_{h+1}}\phi(x,b(x))\rho(x,b(x))\big{(}v(x,b(x))-u(x,b(x))b^{\prime}(x)\big{)}{\rm
d}x$
$\displaystyle\quad+\sum_{h,i}\int_{W_{h,i}}\big{(}s_{h,i}(\rho^{+}u^{+}-\rho^{-}u^{-})-(\rho^{+}v^{+}-\rho^{-}v^{-})\big{)}\phi\,{\rm
d}x$ $\displaystyle=:\text{IV}+\text{V}+\text{VI},$

where $W_{h,i}=\\{(x,y)\,:\,y=w_{h,i}(x)=s_{h,i}(x-x_{h})+y_{n}(h)\,\text{ for
some }n\\}$ are shock waves or upper/lower edges of rarefaction waves lying in
$\Omega_{\Delta x,\vartheta,h}$, and $\rho^{\pm}=\rho(x,w_{i,h}(x)\pm)$,
$u^{\pm}=u(x,w_{i,h}(x)\pm)$, and $v^{\pm}=v(x,w_{i,h}(x)\pm)$.

We now show

Proposition 6.2. There exists a subsequence of $\\{(u_{m},v_{m})\\}$ such that
IV$\rightarrow 0$ as $m\rightarrow\infty$.

Proof. The first term on the right hand of (6.7) can be rewritten as

$\displaystyle\text{IV}=\sum_{h\geq 1}V_{h}$

with

$\displaystyle V_{h}$
$\displaystyle=\sum_{n=n_{\chi,h}+1}^{n_{b,h}-1}\int_{y_{n-1}(h)}^{y_{n}(h)}\phi(x_{h},y)\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ $\displaystyle\quad+\int_{\chi_{\Delta
x,\vartheta}(x_{h})}^{y_{n_{\chi,h}}(h)+1}\phi(x_{h},y)\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ $\displaystyle\quad+\int_{y_{n_{b,h}}(h)-1}^{b_{\Delta
x,\vartheta}(x_{h})}\phi(x_{h},y)\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y.$

To show $\text{IV}\rightarrow 0$ for some subsequence $\\{(u_{m},v_{m})\\}$,
we now introduce

$\displaystyle\widetilde{V}=\sum_{h\geq 1}\widetilde{V}_{h}$

with

$\displaystyle\widetilde{V}_{h}$
$\displaystyle=\sum_{n=n_{\chi,h}+1}^{n_{b,h}-1}\int_{y_{n-1}(h)}^{y_{n}(h)}\phi(x_{h},y_{n}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ $\displaystyle\quad+\int_{y_{n_{b,h}}(h)}^{b_{\Delta
x,\vartheta}(x_{h})}\phi(x_{h},y_{n_{b,h}+1}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi(x_{h},y_{n_{b,h}}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{\chi,h}}(h)}^{y_{n_{\chi,h}+1}(h)}\phi(x_{h},y_{n_{\chi,h}+1}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ $\displaystyle\quad+\int_{\chi_{\Delta
x,\vartheta}(x_{h})}^{y_{n_{\chi,h}}(h)}\phi(x_{h},y_{n_{\chi,h}}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle=:\sum_{n=n_{\chi,h}+1}^{n_{b,h}-1}\int_{y_{n-1}(h)}^{y_{n}(h)}\phi(x_{h},y_{n}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\check{V}_{h}^{(0)}+\check{V}_{h}^{(1)}+\hat{V}_{h}^{(1)}+\hat{V}_{h}^{(0)}.$

From the construction of approximate solutions, we have

$\displaystyle\check{V}_{h}^{(0)}$ $\displaystyle=O(1)\Delta
x\big{(}|\alpha_{r,2}|+|\omega_{h+1}|+|\Delta\sigma_{b_{h}}|+Q(\Lambda_{b})\big{)}$
(see Case 4.1), $\displaystyle\hat{V}_{h}^{(0)}$ $\displaystyle=O(1)\Delta
x\big{(}|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|+Q(\Lambda_{s})\big{)}$ (see
Case 4.2).

From Proposition 5, we obtain

$\sum_{h=1}^{\infty}\big{(}\check{V}_{h}^{(0)}+\hat{V}_{h}^{(0)}\big{)}=O(1)\Delta
xF(I_{1}).$ (6.8)

Then we write $\check{V}_{h}^{(1)}$ and $\hat{V}_{h}^{(1)}$ as

$\displaystyle\check{V}_{h}^{(1)}$
$\displaystyle=\int_{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi(x_{h},y_{n_{b,h}}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\check{\rho}(x_{h}+,y)\check{u}(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi(x_{h},y_{n_{b,h}}(h))\big{(}\check{\rho}(x_{h}+,y)\check{u}(x_{h}+,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$

and

$\displaystyle\hat{V}_{h}^{(1)}$
$\displaystyle=\int_{y_{n_{\chi,h}}(h)}^{y_{n_{\chi,h}+1}(h)}\phi(x_{h},y_{n_{\chi,h}}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\hat{\rho}(x_{h}+,y)\hat{u}(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{\chi,h}}(h)}^{y_{n_{\chi,h}+1}(h)}\phi(x_{h},y_{n_{\chi,h}}(h))\big{(}\hat{\rho}(x_{h}+,y)\hat{u}(x_{h}+,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y,$

where

$\displaystyle\check{U}(x_{h}+,y)=\tilde{U}(\dfrac{y}{x_{h}};\dfrac{r_{h,n_{b,h}-1}}{x_{h}},U(x_{h}+,r_{h,n_{b,h}-1})),$
$\displaystyle\hat{U}(x_{h}+,y)=\tilde{U}(\dfrac{y}{x_{h}};\dfrac{r_{h,n_{\chi,h}-1}}{x_{h}},U(x_{h}+,r_{h,n_{\chi,h}-1})),$

and $\check{\rho}$ and $\hat{\rho}$ are determined via Bernoulli’s equation.
By the construction of the approximate solutions near the boundary and near
the leading shock, we have

$\displaystyle\int_{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi\big{(}x_{h},y_{n_{b,h}}(h)\big{)}\big{(}\check{\rho}(x_{h}+,y)\check{u}(x_{h}+,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}{\rm
d}y$ $\displaystyle\,\,\,=O(1)\Delta
x\big{(}|\alpha_{r,2}|+|\omega_{h+1}|+|\Delta\sigma_{b_{h}}|+Q(\Lambda_{b})\big{)}$
(see Case 4.1),
$\displaystyle\int_{y_{n_{\chi,h}}(h)}^{y_{n_{\chi,h}+1}(h)}\phi\big{(}x_{h},y_{n_{\chi,h}}(h)\big{)}\big{(}\hat{\rho}(x_{h}+,y)\hat{u}(x_{h}+,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}{\rm
d}y$ $\displaystyle\,\,\,=O(1)\Delta
x\big{(}|\alpha_{l,1}|+|\Delta\sigma_{\chi_{h}}|+Q(\Lambda_{s})\big{)}$ (see
Case 4.2).

Similarly, by Proposition 5, we conclude

$\displaystyle\sum_{h=1}^{\infty}\int_{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi(x_{h},y_{n_{b,h}}(h))\big{(}\check{\rho}(x_{h}+,y)\check{u}(x_{h}+,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y=O(1)\Delta xF(I_{1}),$ (6.9)
$\displaystyle\sum_{h=1}^{\infty}\int_{y_{n_{\chi,h}}(h)}^{y_{n_{\chi,h}+1}(h)}\phi(x_{h},y_{n_{\chi,h}}(h))\big{(}\hat{\rho}(x_{h}+,y)\hat{u}(x_{h}+,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y=O(1)\Delta xF(I_{1}).$

Setting

$\displaystyle\bar{V}_{h}$
$\displaystyle=\sum_{n=n_{\chi,h}+1}^{n_{b,h}-1}\int_{y_{n-1}(h)}^{y_{n}(h)}\phi(x_{h},y_{n}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\phi(x_{h},y_{n_{b,h}}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\check{\rho}(x_{h}+,y)\check{u}(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{\chi,h}}(h)}^{y_{n_{\chi,h}+1}(h)}\phi(x_{h},y_{n_{\chi,h}}(h))\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\hat{\rho}(x_{h}+,y)\hat{u}(x_{h}+,y)\big{)}\,{\rm
d}y.$

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

$H=\prod_{h=0}^{\infty}[0,1)=\\{\vartheta=(\vartheta_{0},\vartheta_{1},\vartheta_{2},\dots,\vartheta_{h},\dots)\,:\,\vartheta_{h}\in[0,1),\
h=0,1,2,\cdots\\}.$

Denoting $\bar{y}=y_{n-1}(h)+\vartheta_{h}\big{(}y_{n}(h)-y_{n-1}(h)\big{)}$,
we obtain from (4.2) that

$\displaystyle\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)$
$\displaystyle=\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}-,\bar{y})u(x_{h}-,\bar{y})+\rho(x_{h}+,\bar{y})u(x_{h}+,\bar{y})-\rho(x_{h}+,y)u(x_{h}+,y)$
$\displaystyle=O(1)|\alpha|+O(1)|\alpha||\Delta\sigma_{\alpha}|+O(1)|\Delta\sigma_{\alpha}|$
$\displaystyle=O(1)(|\alpha|+|\Delta\sigma_{\alpha}|),$

where $\alpha$ is an elementary wave in $\Omega_{\Delta x,\vartheta,h,j}$ and
$\Delta\sigma_{\alpha}$ is the change of the $\sigma-$coordinate in the
elementary wave $\alpha$. Denote the elementary waves in $\Omega_{\Delta
x,\vartheta,h}$ by $\alpha_{h,i}$. Then we

$\bar{V}_{h}=O(1)\Big{(}\sum_{i\leq
0}|\alpha_{h,i}|+\sigma^{*}-\sigma_{*}\Big{)}\Delta x,$ (6.10)

which implies

$\displaystyle\sum_{h\geq 1}\int_{H}\bar{V}_{h}^{2}{\rm
d}\vartheta=O(1)\text{diam}(\text{supp}\phi)\Big{(}\sum_{i\leq
0}|\alpha_{h,i}|+\sigma^{*}-\sigma_{*}\Big{)}^{2}\Delta x.$

Next, we need the following lemma.

Lemma 6.1. The approximate solutions $\\{U_{\Delta x,\vartheta}(x,y)\\}$
satisfy

$\int_{0}^{1}\int_{y_{n-1}(h)}^{y_{n}(h)}\big{(}U_{\Delta
x,\vartheta}(x_{h}-,y)-U_{\Delta x,\vartheta}(x_{h}+,y)\big{)}\,{\rm d}y{\rm
d}\vartheta_{h}=O(1)(\Delta x)^{3}+O(1)(|\alpha|+|\beta|)(\Delta x)^{2}.$
(6.11)

Proof. We now give a proof when $\alpha$ and $\beta$ are both shock waves,
since the remaining cases can be obtained similarly.

Suppose that $\alpha$ and $\beta$ issue from $(x_{h-1},y_{n-1}(h-1))$ and
$(x_{h-1},y_{n}(h-1))$, and end at $(x_{h},r_{1})$ and $(x_{h},r_{2})$,
respectively. Set $a_{1}=\frac{r_{1}-y_{n-1}(h)}{y_{n}(h)-y_{n-1}(h)}$ and
$a_{2}=\frac{r_{2}-y_{n-1}(h)}{y_{n}(h)-y_{n-1}(h)}$. From the construction of
approximate solutions, we have

$\displaystyle\int_{0}^{1}\int_{y_{n-1}(h)}^{y_{n}(h)}\big{(}U_{\Delta
x,\vartheta}(x_{h}-,y)-U_{\Delta x,\vartheta}(x_{h}+,y)\big{)}\,{\rm d}y{\rm
d}\vartheta_{h}$
$\displaystyle=\int_{0}^{1}\int_{y_{n-1}(h)}^{y_{n}(h)}U_{\Delta
x,\vartheta}(x_{h}-,y)\,{\rm d}y{\rm
d}\vartheta_{h}-\int_{0}^{1}\int_{y_{n-1}(h)}^{y_{n}(h)}U_{\Delta
x,\vartheta}(x_{h}+,y)\,{\rm d}y{\rm d}\vartheta_{h}$
$\displaystyle=\int_{y_{n-1}(h)}^{r_{1}}\tilde{U}(\frac{y}{x_{h}};\sigma_{2},U_{l})\,{\rm
d}y+\int_{r_{1}}^{r_{2}}\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))\,{\rm
d}y$
$\displaystyle\quad+\int_{r_{2}}^{y_{n}(h)}\tilde{U}(\frac{y}{x_{h}};\sigma_{1},\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))))\,{\rm
d}y-a_{1}\int_{y_{n-1}(h)}^{y_{n}(h)}\tilde{U}(\frac{y}{x_{h}};\sigma_{2},U_{l})\,{\rm
d}y$
$\displaystyle\quad-(a_{2}-a_{1})\int_{y_{n-1}(h)}^{y_{n}(h)}\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))\,{\rm
d}y$
$\displaystyle\quad-(1-a_{2})\int_{y_{n-1}(h)}^{y_{n}(h)}\tilde{U}(\frac{y}{x_{h}};\sigma_{1},\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))))\,{\rm
d}y.$

Since
$\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))=\tilde{U}(\frac{y}{x_{h}};\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l})))$,
we obtain

$\displaystyle\int_{0}^{1}\int_{y_{n-1}(h)}^{y_{n}(h)}(U_{\Delta
x,\vartheta}(x_{h}-,y)-U_{\Delta x,\vartheta}(x_{h}+,y))\,{\rm d}y{\rm
d}\vartheta_{h}$
$\displaystyle=\int_{y_{n-1}(h)}^{r_{1}}(1-a_{1})\big{(}\tilde{U}(y/x_{h};\sigma_{2},U_{l})-\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))\big{)}\,{\rm
d}y$
$\displaystyle\quad-\int_{y_{n-1}(h)}^{r_{1}}(1-a_{2})\big{(}\tilde{U}(\frac{y}{x_{h}};\sigma_{1},\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))))-\tilde{U}(\frac{y}{x_{h}};\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l})))\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{r_{1}}^{r_{2}}a_{1}\big{(}\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))-\tilde{U}(\frac{y}{x_{h}};\sigma_{2},U_{l})\big{)}\,{\rm
d}y$
$\displaystyle\quad-\int_{r_{1}}^{r_{2}}(1-a_{2})\big{(}\tilde{U}(\frac{y}{x_{h}};\sigma_{1},\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))))-\tilde{U}(\frac{y}{x_{h}};\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l})))\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{r_{2}}^{y_{n}(h)}a_{2}\big{(}\tilde{U}(\frac{y}{x_{h}};\sigma_{1},\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))))-\tilde{U}(\frac{y}{x_{h}};\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l})))\big{)}\,{\rm
d}y$
$\displaystyle\quad-\int_{r_{2}}^{y_{n}(h)}a_{1}\big{(}\tilde{U}(\frac{y}{x_{h}};\sigma_{2},U_{l})-\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))\big{)}\,{\rm
d}y.$

Then, by Taylor’s expansion, we have

$\displaystyle\tilde{U}(\frac{y}{x_{h}};\sigma_{2},U_{l})-\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))$
$\displaystyle\,\,\,=U_{l}-\Phi(0,\alpha_{2};U_{l})+A_{1}(y-y_{n-1}(h))+O(1)(y-y_{n-1}(h))^{2},$
$\displaystyle\tilde{U}(\frac{y}{x_{h}};\sigma_{1},\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))))-\tilde{U}(\frac{y}{x_{h}};\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l})))$
$\displaystyle\,\,\,=\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l})))-\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))+A_{2}(y-y_{n}(h))+O(1)(y-y_{n}(h))^{2},$

with

$\displaystyle A_{1}=$
$\displaystyle\left.\partial_{y}\Big{(}\tilde{U}(\frac{y}{x_{h}};\sigma_{2},U_{l})-\tilde{U}(\frac{y}{x_{h}};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))\Big{)}\right|_{y=y_{n-1}(h)},$
$\displaystyle A_{2}=$
$\displaystyle\left.\partial_{y}\Big{(}\tilde{U}(\frac{y}{x_{h}};\sigma_{1},\Phi(\beta_{1},0;\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l}))))-\tilde{U}(\frac{y}{x_{h}};\tilde{U}(\sigma_{1};\sigma_{2},\Phi(0,\alpha_{2};U_{l})))\Big{)}\right|_{y=y_{n}(h)}.$

A direct computation leads to

$\displaystyle\int_{0}^{1}\int_{y_{n-1}(h)}^{y_{n}(h)}\big{(}U_{\Delta
x,\vartheta}(x_{h}-,y)-U_{\Delta x,\vartheta}(x_{h}+,y)\big{)}\,{\rm d}y{\rm
d}\vartheta_{h}$
$\displaystyle=O(1)(y_{n}(h)-y_{n-1}(h))^{3}+\dfrac{1}{2}A_{1}(r_{1}-y_{n-1}(h))(r_{1}-y_{n}(h))-\dfrac{1}{2}A_{2}(r_{2}-y_{n-1}(h))(r_{2}-y_{n}(h)).$

Noting that $A_{1}=O(1)|\alpha|$ and $A_{2}=O(1)|\beta|$, together with the
Courant-Friedrichs-Lewy condition, we conclude (6.11). $\square$

Substituting $U_{\Delta x,\vartheta}$ in Lemma 6 by $\rho u$ and carrying out
the same process lead to

$\displaystyle\int_{0}^{1}\bar{V}_{h}{\rm
d}\vartheta_{h}=O(1)\Big{(}\text{diam(supp$\phi$)}+\sum_{i\leq
0}|\alpha_{h,i}|\Big{)}(\Delta x)^{2}=O(1)(\Delta x)^{2}.$

As in (6.10), we obtain

$\displaystyle\bar{V}_{k}=O(1)\Big{(}\sum_{i\leq
0}|\alpha_{k,i}|+\sigma^{*}-\sigma_{*}\Big{)}\Delta x.$

Then

$\displaystyle\sum_{h>k}\int_{H}\bar{V}_{h}\bar{V}_{k}\,{\rm d}\vartheta\leq$
$\displaystyle\sum_{h>k}\left|\int_{0}^{1}\bar{V}_{h}d\vartheta_{h}\right|\int_{0}^{1}|\bar{V}_{k}|\,{\rm
d}\hat{\vartheta}_{h}=O(1)\big{(}\text{diam(supp$\phi$)}\big{)}^{2}\Delta x,$

where ${\rm d}\hat{\vartheta}_{h}={\rm d}\vartheta_{0}\cdots{\rm
d}\vartheta_{h-1}{\rm d}\vartheta_{h+1}\cdots$.

Since

$\displaystyle\|\bar{V}\|_{L^{2}(H)}=\sum_{h\geq 1}\int_{H}\bar{V}_{h}^{2}{\rm
d}\vartheta+2\sum_{h>k}\int_{H}\bar{V}_{h}\bar{V}_{k}\,{\rm d}\vartheta,$

we conclude that

$\displaystyle\|\bar{V}\|_{L^{2}(H)}\rightarrow 0\qquad\text{as }\Delta
x\rightarrow 0,$

which, combining with (6.8)–(6.9), gives a subsequence (still denoted by)
$\\{(u_{m},v_{m})\\}$ such that $\widetilde{V}\rightarrow 0$ almost
everywhere. Meanwhile, we have

$\displaystyle V_{h}-\widetilde{V}_{h}$
$\displaystyle=\sum_{n=n_{\chi,h}+1}^{n_{b,h}-1}\int_{y_{n-1}(h)}^{y_{n}(h)}\big{(}\phi(x_{h},y_{n}(h))-\phi(x_{h},y)\big{)}\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{b,h}-1}(h)}^{y_{n_{b,h}}(h)}\big{(}\phi(x_{h},y_{n_{b,h}}(h))-\phi(x_{h},y)\big{)}\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ $\displaystyle\quad+\int_{y_{n_{b,h}}(h)}^{b_{\Delta
x,\vartheta}(x_{h})}\big{(}\phi(x_{h},y_{n_{b,h}+1}(h))-\phi(x_{h},y)\big{)}\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ $\displaystyle\quad+\int_{\chi_{\Delta
x,\vartheta}(x_{h})}^{y_{n_{\chi,h}}(h)}\big{(}\phi(x_{h},y_{n_{\chi,h}}(h))-\phi(x_{h},y)\big{)}\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$
$\displaystyle\quad+\int_{y_{n_{\chi,h}}(h)}^{y_{n_{\chi,h}+1}(h)}\big{(}\phi(x_{h},y_{n_{\chi,h}+1}(h))-\phi(x_{h},y)\big{)}\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y$ $\displaystyle=O(1)\Delta
x\sum_{n=n_{\chi,h}+1}^{n_{b,h}-1}\int_{y_{n-1}(h)}^{y_{n}(h)}\big{(}\rho(x_{h}-,y)u(x_{h}-,y)-\rho(x_{h}+,y)u(x_{h}+,y)\big{)}\,{\rm
d}y+O(1)(\Delta x)^{2}$ $\displaystyle=O(1)\Big{(}\sum_{i\leq
0}|\alpha_{h,i}|+\sigma^{*}-\sigma_{*}+1\Big{)}(\Delta x)^{2},$

which leads to

$\displaystyle\text{IV}-\widetilde{V}=$ $\displaystyle\sum_{h\geq
1}V_{h}-\widetilde{V}_{h}=O(1)\,\text{diam}(\text{supp}\phi)\Delta x.$

Thus, $\text{IV}\rightarrow 0$ as $m\rightarrow\infty$ for some subsequence
$\\{(u_{m},v_{m})\\}$. $\square$

Proposition 6.3. V, VI$\rightarrow 0$ as $\Delta x\rightarrow 0$.

Proof. Since

$\displaystyle
b^{\prime}(x)=\frac{v\big{(}x_{h}+,b(x_{h})-\big{)}}{u\big{(}x_{h}+,b(x_{h})-\big{)}}\qquad\text{for
}x\in(x_{h},x_{h+1}),$

it follows from the construction of our approximate solution that

$\displaystyle
v\big{(}x,b(x)\big{)}-u\big{(}x,b(x)\big{)}b^{\prime}(x)=O(1)\Delta x.$

Therefore, we have

$\displaystyle\text{V}=O(1)\,\text{diam}(\text{supp}\phi)\,\Delta x\rightarrow
0\qquad\text{as }\Delta x\rightarrow 0.$

As for VI, we divide this term into three parts. The first part is the
integral along the leading shock, where $W_{h,i}=S_{\Delta x,\vartheta,h}$.
For this part, by similar arguments in treating V, we have

$\displaystyle\sum_{h}\int_{S_{\Delta
x,\vartheta,h}}[s_{h}(\rho^{+}u^{+}-\rho^{-}u^{-})-(\rho^{+}v^{+}-\rho^{-}v^{-})]\phi\,{\rm
d}x=O(1)\Delta x.$

The second part is the integral along the upper or lower edges of rarefaction
waves and therefore vanishes automatically. The third part is the integral
along the weak shock waves, that is, $W_{h,i}\neq S_{\Delta x,\vartheta,h}$.
In this case, by (4.1), we have

$\displaystyle\rho^{+}u^{+}-\rho^{-}u^{-}=(\rho^{+}u^{+}-\rho^{-}u^{-})|_{x=x_{h}+}+O(1)(\rho^{+}u^{+}-\rho^{-}u^{-})|_{x=x_{h}+}\Delta
x,$
$\displaystyle\rho^{+}v^{+}-\rho^{-}v^{-}=(\rho^{+}v^{+}-\rho^{-}v^{-})|_{x=x_{h}+}+O(1)(\rho^{+}v^{+}-\rho^{-}v^{-})|_{x=x_{h}+}\Delta
x.$

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

$\displaystyle\sum_{i}s_{h,i}(\rho^{+}u^{+}-\rho^{-}u^{-})-(\rho^{+}v^{+}-\rho^{-}v^{-})=O(1)\sum_{i}|\alpha_{S,h,i}|\Delta
x,$

where $\alpha_{S,h,i}$ are the weak shock waves in $\Omega_{\Delta
x,\vartheta,h}$. Combining all the three parts together, we have

$\displaystyle\text{IV}=O(1)\,\text{diam}(\text{supp}\phi)\,\sum_{i}|\alpha_{S,h,i}|\Delta
x+O(1)\Delta x.$

By Proposition 5, $\sum_{i}|\alpha_{S,h,i}|$ is uniformly bounded with respect
to $h$. Therefore, $\text{IV}\rightarrow 0$ as $\Delta x\rightarrow 0$.
$\square$

With all the arguments stated above, a standard procedure as in [18, 38] gives
the following main theorem.

Theorem 6.1. Suppose that (1.2)–(1.2), $1<\gamma<3$, and $0<p_{0}<p^{*}$.
Then, when $M_{\infty}$ is sufficiently large, there are $\epsilon_{0}>0$ and
a null set $\mathcal{N}$ such that, if T.V. $p^{b}<\epsilon_{0}$, for each
$\vartheta\in\prod_{h=0}^{\infty}[0,1)\backslash\mathcal{N}$, there exist both
a subsequence $\\{\Delta_{i}\\}_{i=0}^{\infty}\subset\\{\Delta x\\}$ of the
mesh size with $\Delta_{i}\rightarrow 0$ as $i\rightarrow\infty$ and a triple
of functions $b_{\vartheta}(x)$ with $b_{\vartheta}(0)=0$,
$\chi_{\vartheta}(x)$ with $\chi_{\vartheta}(0)=0$, and $U_{\vartheta}(x,y)\in
O_{\varepsilon_{0}}\big{(}G(s_{0})\cap W(p_{0},p_{\infty})\big{)}$ such that

  * (i)

$b_{\Delta_{i},\vartheta}(x)$ converges to $b_{\vartheta}(x)$ uniformly in any
bounded $x$-interval;

  * (ii)

$\chi_{\Delta_{i},\vartheta}(x)$ converges to $\chi_{\vartheta}(x)$ uniformly
in any bounded $x$-interval;

  * (iii)

$b^{\prime}_{\Delta_{i},\vartheta}(x)$ converges to
$b^{\prime}_{\vartheta}(x)\in BV([0,\infty))$ a.e. and $\dfrac{{\rm
d}b_{\vartheta}(x)}{{\rm d}x}=b^{\prime}(x)$, a.e. with
$|b^{\prime}_{\vartheta}-b_{0}|<C\epsilon_{0}$;

  * (iv)

$s_{\Delta_{i},\vartheta}(x)$ converges to $s_{\vartheta}(x)\in
BV([0,\infty))$ a.e. and $\dfrac{{\rm d}\chi_{\vartheta}(x)}{{\rm
d}x}=\sigma_{\vartheta}(x)$, a.e. with $|s_{\vartheta}-s_{0}|<C\epsilon_{0}$;

  * (v)

$U_{\Delta_{i},\vartheta}(x,\cdot)$ converges to
$U_{\vartheta}\in\textbf{L}^{1}_{\text{loc}}(-\infty,b_{\vartheta}(x))$ for
every $x>0$, and $U_{\vartheta}$ is a global entropy solution of the inverse
problem (1.1)–(1.2) and satisfies (1.8)–(1.9).

##  7\. Asymptotic Behavior of Global Entropy Solutions

To establish the asymptotic behavior of global entropy solutions, we need