#  $L^{1}$–Stability of Vortex Sheets and Entropy Waves
in Steady Supersonic Euler Flows over Lipschitz Walls

Gui-Qiang Chen Vaibhav Kukreja  Gui-Qiang G. Chen, Mathematical Institute,
University of Oxford, Oxford, OX1 3LB, UK; Department of Mathematics,
Northwestern University, Evanston, IL 60208, USA; School of Mathematical
Sciences, Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>Vaibhav Kukreja, Instituto de Matem$\acute{\text{a}}$tica Pura e Aplicada
(IMPA), Rio de Janeiro, Brazil; Department of Mathematics, Northwestern
University, Evanston, IL 60208, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We establish the well-posedness of compressible vortex sheets and entropy
waves in two-dimensional steady supersonic Euler flows over Lipschitz walls
under a $BV$ boundary perturbation. In particular, when the total variation of
the incoming flow perturbation around the background strong vortex
sheet/entropy wave is small, we prove that the two-dimensional steady
supersonic Euler flows containing a strong vortex sheet/entropy wave past a
Lipschitz wall are $L^{1}$–stable. Both the Lipschitz wall (whose boundary
slope function has small total variation) and incoming flow perturb the
background strong vortex sheet/entropy wave. The weak waves are reflected
after nonlinear waves interact with the strong vortex sheet/entropy wave and
the wall boundary. Using the wave-front tracking method, the existence of
solutions in $BV$ over Lipschitz walls is first shown, when the incoming flow
perturbation of the background strong vortex sheet/entropy wave has small
total variation. Then we establish the $L^{1}$–contraction of the solutions
with respect to the incoming flows. To achieve this, a Lyapunov functional,
equivalent to the $L^{1}$–distance between two solutions containing strong
vortex sheets/entropy waves, is carefully constructed to include the nonlinear
waves generated both by the wall boundary and from the incoming flow. This
functional is then shown to decrease in the flow direction, leading to the
$L^{1}$–stability, as well as the uniqueness, of the solutions. Furthermore,
the uniqueness of solutions extends to a larger class of viscosity solutions.

###### Key words and phrases:

Full Euler equations, entropy waves, compressible vortex sheets,
$L^{1}$–stability, steady flows, supersonic Euler flow, Riemann solutions,
Lipschitz wall, $BV$ perturbation, Glimm’s functional, nonlinear interaction,
global existence

###### 2010 Mathematics Subject Classification:

Primary: 35B35, 35B40, 76J20, 35L65, 85A05, 35A05

##  1\. Introduction

We study the well-posedness of two-dimensional steady supersonic Euler flows
past a curved Lipschitz wall containing strong vortex sheets/entropy waves in
the $L^{1}$–norm. The inviscid compressible flows are governed by the two-
dimensional steady Euler system:

$\begin{cases}(\rho u)_{x}+(\rho v)_{y}=0,\\\ (\rho u^{2}+p)_{x}+(\rho
uv)_{y}=0,\\\ (\rho uv)_{x}+(\rho v^{2}+p)_{y}=0,\\\ (\rho
u(E+\frac{p}{\rho}))_{x}+(\rho v(E+\frac{p}{\rho}))_{y}=0,\end{cases}$ (1.1)

with $(u,v)$, $p$, $\rho$, and $E$ representing the fluid velocity, scalar
pressure, density, and total energy, respectively. Furthermore, the total
energy $E$ is explicitly given by

$E=\frac{1}{2}(u^{2}+v^{2})+e(\rho,p),$

where the internal energy $e$ can be written as a function of $(\rho,p)$
defined through the thermodynamical relations. The temperature $T$ and entropy
$S$ are the other two thermodynamic variables.

In the case of an ideal gas, the pressure p and internal energy e can be
expressed as

$p=R\rho T,\hskip 14.22636pte=c_{\nu}T$ (1.2)

with the adiabatic index $\gamma$ given by

$\gamma=1+\frac{R}{c_{\nu}}>1.$ (1.3)

In particular, in terms of the density $\rho$ and entropy S, we have

$p=p(\rho,S)=\kappa\rho^{\gamma}e^{S/c_{\nu}},\qquad
e=\frac{\kappa}{\gamma-1}\rho^{\gamma-1}e^{S/c_{\nu}}=\frac{RT}{\gamma-1},$
(1.4)

The constants $R,c_{\nu}$, and $\kappa$ in the above relations are all greater
than zero.

When the entropy $S$ = constant, the flow is called isentropic. In this case,
the pressure $p$ can be written as a function of the density $\rho$,
$p=p(\rho)$, and the flow is governed by the isentropic Euler equations:

$\begin{cases}(\rho u)_{x}+(\rho v)_{y}=0,\\\ (\rho u^{2}+p)_{x}+(\rho
uv)_{y}=0,\\\ (\rho uv)_{x}+(\rho v^{2}+p)_{y}=0.\end{cases}$ (1.5)

Then, by scaling, the pressure-density relation is

$p(\rho)=\frac{\rho^{\gamma}}{\gamma}.$ (1.6)

The adiabatic exponent $\gamma>1$ corresponds to the isentropic polytropic
gas. The limiting case $\gamma=1$ corresponds to the isothermal flow. Define

$c=\sqrt{p_{\rho}(\rho,S)}$

as the sonic speed. For polytropic gases, the sonic speed is $c=\sqrt{\gamma
p/\rho}$.

The flow type is classified by the Mach number
$M=\frac{\sqrt{u^{2}+v^{2}}}{c^{2}}$. When $M>1$, system (1.1) or (1.5)
governs a supersonic flow (i.e., $u^{2}+v^{2}>c^{2}$), which has all real
eigenvalues and is hyperbolic. For $M<1$, system (1.1) or (1.5) governs a
subsonic flow (i.e., $u^{2}+v^{2}<c^{2}$), which has complex eigenvalues and
is elliptic-hyperbolic mixed and composite. When $M=1$, the flow is called
sonic.

We are interested in whether compressible vortex sheets/entropy waves in
supersonic flow over the Lipschitz wall are always stable under the $BV$
perturbation of the incoming flow. Multidimensional steady supersonic Euler
flows are important in many physical applications (cf. Courant-Friedrichs
[12]). In particular, when the upstream flow is a uniform steady flow above
the plane wall in $x<0$ all the time, the flow downstream above a Lipschitz
wall in $x>0$ is governed by a steady Euler flow after a sufficiently long
time. Moreover, compressible vortex sheets and entropy waves occur
ubiquitously in nature and are fundamental waves. Furthermore, since steady
Euler flows are asymptotic states and may be global attractors of the
corresponding unsteady Euler flows, it is important to establish the existence
of steady Euler flows and understand their qualitative behavior to shed light
on the long-time asymptotic behavior of the unsteady compressible Euler flows,
one of the most fundamental problems in mathematical fluid dynamics which is
still wide open.

We observe that the stability of contact discontinuities for the Cauchy
problem for strictly hyperbolic systems in one space dimension under a $BV$
perturbation has been studied by Sabl$\acute{\text{e}}$-Tougeron [25] and
Corli–Sabl$\acute{\text{e}}$-Tougeron [13]. In particular, the reflection
coefficients, such as $K_{11}$ here, are required to be less than one, which
is the stability condition for the mixed problem in the strip $\\{(t,x):t\geq
0,-1<x<1\\}$ in the earlier works; see, e.g., Sabl$\acute{\text{e}}$-Tougeron
[25]. Working with the non-isentropic Euler system (1.1) and a uniform
upstream flow, Chen-Zhang-Zhu [11] first proved the global existence in $BV$
of supersonic Euler flows containing a strong vortex sheet/entropy wave under
the $BV$ perturbation of the Lipschitz wall by using the Glimm scheme. The
essential difference between system (1.1) as analyzed in [11] (and in Sections
2–7 here) and strictly hyperbolic systems as considered in [13, 25] is that
two of the four characteristic eigenvalues coincide and have two linearly
independent eigenvectors which determine precisely the compressible vortex
sheets and entropy waves so that two independent parameters are required to
describe them, respectively.

In this paper, for completeness, we first show, via the wave-front tracking
method, the existence of solutions to the problem when a small $BV$
perturbation is added to the uniform incoming flow. Then the $L^{1}$–stability
of entropy solutions containing strong vortex sheets/entropy waves is
established. As corollaries of these results, the estimates on the uniformly
Lipschitz semigroup $\mathscr{S}$ of entropy solutions generated by the wave-
front tracking approximations are obtained, and the uniqueness of weak
solutions containing strong vortex sheets/entropy waves is established in a
larger set of solutions, namely the class of viscosity solutions.

In the following, we focus mainly on the problem in the region $\Omega$ over
the Lipschitz wall for the supersonic Euler flows $U(u,v,p,\rho)$ governed by
system (1.1), given that the corresponding problem for the isentropic system
(1.5) is simpler to analyze. The subsequent figure provides the schematic
diagram for the problem we study:

Figure 1.1. Stability of the compressible vortex sheet/entropy wave in
supersonic flow

The boundary and initial data in the problem are as follows:

  * (i)

There is a Lipschitz function $g\in{\rm Lip}(\mathbb{R}_{+};\mathbb{R})$ such
that

$g(0)=g^{\prime}(0+)=0,\hskip
8.53581pt\displaystyle\lim_{x\to\infty}\text{arctan}(g^{\prime}(x+))=0,\hskip
8.53581ptg^{\prime}\in BV(\mathbb{R}_{+};\mathbb{R})$

and

${\rm TV}(g^{\prime}(\cdot))<\varepsilon\quad\hskip 14.22636pt\text{for some
constant }\varepsilon>0.$

Denote $\Omega\mathrel{\mathop{:}}=\\{(x,y):y>g(x),x\geq 0\\}$,
$\Gamma\mathrel{\mathop{:}}=\\{(x,y):y=g(x),x\geq 0\\}$, and
$\textbf{n}(x\pm)$ =
$\frac{(-g^{\prime}(x\pm),1)}{\sqrt{(g^{\prime}(x\pm))^{2}+1}}$ as the outer
normal vectors to $\Gamma$ at the respective points $x\pm$ (cf. Fig. 1.1).

  * (ii)

The incoming flow $U=\overline{U}(y):=U_{0}^{b}+\widetilde{U_{0}}$ at $x=0$ is
composed of two parts:

    1. (a)

The upstream flow $U_{0}^{b}$ consists of one straight vortex sheet/entropy
wave $y=y_{0}^{\ast}>0$ and two constant vectors $U^{-}_{0}=U_{-}$, when
$0<y<y_{0}^{\ast}$, and $U^{+}_{0}=U_{+}$, when $y>y_{0}^{\ast}>0$, satisfying

$v_{-}=v_{+}=0,\hskip 14.22636ptu_{\pm}>c_{\pm}>0,$

where $c_{\pm}=\sqrt{\gamma p_{\pm}/\rho_{\pm}}$ is the sonic speed of state
$U_{\pm}$.

    2. (b)

The $BV$ perturbation
$\widetilde{U_{0}}=(\tilde{u}_{0},\tilde{v}_{0},\tilde{p}_{0},\tilde{\rho}_{0})(y)\in
L^{1}\cap BV(\mathbb{R};\mathbb{R}^{4})$ at $x=0$ so that ${\rm
TV}(\widetilde{U_{0}})\ll 1$.

Then we consider the following initial-boundary value problem for system
(1.1):

$\displaystyle\textbf{Cauchy Condition}:\qquad\quad$ $\displaystyle
U|_{x=0}=\overline{U}(y)=U_{0}^{b}+\widetilde{U_{0}};$ (1.7)
$\displaystyle\textbf{Boundary Condition}:\quad\quad$
$\displaystyle(u,v)\cdot\textbf{n}=0\qquad\text{ on }\Gamma.$ (1.8)

Definition 1.1 (Admissible entropy solutions). A $BV$ function $U=U(x,y)$ is
said to be an entropy solution of the initial-boundary value problem (1.1) and
(1.7)–(1.8) if and only if the following conditions hold:

  * (i)

$U$ is a weak solution of (2.1) and satisfies

$U|_{x=0}=\overline{U}(y)\quad\mbox{and}\quad(u,v)\cdot\textbf{n}|_{y=g(x)}=0\,\,\,\text{
in the trace sense;}$

  * (ii)

$U$ satisfies the steady entropy Clausius inequality:

$(\rho uS)_{x}+(\rho vS)_{y}\geq 0$ (1.9)

in the distributional sense in $\Omega$ including the Lipschitz wall boundary.

One of the essential developments within this paper is to develop suitable
methods to deal with the challenges caused by the nonstrictly hyperbolicity of
the system and the Lipschitz wall boundary, in comparison with the previous
progress with the strictly hyperbolic systems of conservation laws,
particularly to the analysis of the Cauchy problem. For supersonic Euler flow
with a strong shock-front emanating from the wedge vertex, Chen-Li [10] worked
out the issue for a Lipschitz wedge boundary. We now discuss some main
differences in our work here from the Cauchy problem and the resulting key
difficulties. We remark that, in the case of the Cauchy problem concerning
only $weak$ waves, the decrease of the Lyapunov functional and the
$L^{1}$–stability of the solutions were obtained through the cancellation of
distances on both sides of waves. In the presence of a strong shock, for the
$L^{1}$–stability of solutions of the Cauchy problem for strictly hyperbolic
systems of conservation laws, the Lyapunov functional was found to decrease by
employing the strength of the strong shock to control the strengths of weak
waves of the other families (e.g., see Lewicka-Trivisa [22]). In contrast with
our Lipschitz wall problem, which is a problem of initial-boundary value type,
there is no such cancellation by the boundary as only one-side is possible
near it. Furthermore, no strong vortex sheets/entropy waves (characteristic
discontinuities) nor strong shocks are present to handle the strength of the
weak waves of the other families, and the terms in the estimates for the first
and fourth family carry different signs. As such, it is difficult to say
whether the functional can be made to decrease for our case of strong vortex
sheets and entropy waves with multiplicity of eigenvalues. One of the key
steps to resolve this is to use the physical feature of the boundary condition
that the flow of two solutions near the boundary must run in parallel (also
see [10]). This observation helps us to obtain additional quantitative
relations near the boundary. Then, applying suitable weights and adjustments
in the coefficients of the Lyapunov functional and using the cancellation
between the different families, the functional is found to decrease in the
flow direction.

The rest of the paper is organized as follows. In Section 2, we recall some
fundamental properties of the two-dimensional steady Euler system (1.1) and
discuss related nonlinear waves and wave interaction estimates. In Section 3,
the wave-front tracking algorithm is discussed, working in the presence of
strong vortex sheets/entropy waves, the suitable interaction potential
$\mathcal{Q}$ is constructed, including the effect of the Lipschitz wall, and
the existence of entropy solutions in $BV$ is established for the initial-
boundary value problem. In Section 4, we construct the Lyapunov functional
$\mathit{\Phi}$ (equivalent to the $L^{1}$–distance between two entropy
solutions $U$ and $V$) to include the nonlinear waves produced by the wall
boundary vertices. Then, in Section 5, the monotone decrease of the functional
$\mathit{\Phi}$ is established in the flow direction, leading to the
$L^{1}$–stability of the solutions containing strong vortex sheets/entropy
waves. Using the estimates established in Sections 3–5, in Section 6, we
obtain the existence of a Lipschitz semigroup of solutions generated by a
wave-front tracking approximation, as well as some estimates on the uniformly
Lipschitz semigroup $\mathscr{S}$ produced by the limit of wave-front tracking
approximations. Moreover, the uniqueness of solutions with strong vortex
sheets/entropy waves is obtained in the larger class of viscosity solutions.

##  2\. Adiabatic Euler equations: Nonlinear waves and wave interactions

In this section, we first present some basic properties of the steady Euler
system (1.1). Then related nonlinear waves and interaction estimates are
discussed, which will be employed in the later sections.

Consider the following vector functions of the solution $U$:

$W(U)=(\rho u,\rho u^{2}+p,\rho uv,\rho u(h+\frac{u^{2}+v^{2}}{2}))^{\top},$
$H(U)=(\rho v,\rho uv,\rho v^{2}+p,\rho v(h+\frac{u^{2}+v^{2}}{2}))^{\top},$

where $h=\frac{\gamma p}{(\gamma-1)\rho}$. Then the steady Euler equations in
(1.1) can be expressed in the following conservative form:

$W(U)_{x}+H(U)_{y}=0,\hskip 14.22636ptU=(u,v,p,\rho)^{\top}$ (2.1)

When $U(x,y)$ is a smooth solution, system (2.1) is equivalent to

$\nabla_{U}W(U)U_{x}+\nabla_{U}H(U)U_{y}=0.$ (2.2)

Then the roots of the fourth degree polynomial

${\rm det}(\lambda\nabla_{U}W(U)-\nabla_{U}H(U)),$ (2.3)

are the eigenvalues of (2.1); that is, the solutions of the equation

$(v-\lambda u)^{2}\big{(}(v-\lambda u)^{2}-c^{2}(1+\lambda^{2})\big{)}=0,$
(2.4)

where $c=\sqrt{\frac{\gamma p}{\rho}}$ is the sonic speed. For supersonic
flows (i.e. $u^{2}+v^{2}>c^{2}$), system (2.1) is hyperbolic. Specifically,
when $u>c$, system (2.1) has four real eigenvalues in the $x$-direction:

$\displaystyle\lambda_{d}$ $\displaystyle=$
$\displaystyle\frac{uv+(-1)^{d}c\sqrt{u^{2}+v^{2}-c^{2}}}{u^{2}-c^{2}},\qquad
d=1,4;$ $\displaystyle\lambda_{k}$ $\displaystyle=$
$\displaystyle\frac{v}{u},\qquad k=2,3,$ (2.5)

with the four corresponding linearly independent eigenvectors given by

$\displaystyle\textbf{r}_{d}$ $\displaystyle=$
$\displaystyle\kappa_{d}(-\lambda_{d},1,\rho(\lambda_{d}u-v),\frac{\rho(\lambda_{d}u-v)}{c^{2}})^{\top},\qquad
d=1,4,$ $\displaystyle\textbf{r}_{2}$ $\displaystyle=$
$\displaystyle(u,v,0,0)^{\top},\hskip
14.22636pt\textbf{r}_{3}=(0,0,0,\rho)^{\top},$ (2.6)

where $\kappa_{d}$ the re-normalization factors such that
$\textbf{r}_{d}\cdot\nabla\lambda_{d}=1$, given that the $d$th-characteristic
fields, $d=1,4$, are genuinely nonlinear. The second and third linearly
degenerate characteristic fields satisfy
$\textbf{r}_{k}\cdot\nabla\lambda_{k}=0$, $k=2,3$, which correspond to vortex
sheets and entropy waves, respectively.

The wave curves in the phase space are now described. The Rankine-Hugoniot
jump conditions for (2.1) are

$\sigma\left[W(u)\right]=\left[H(u)\right],$ (2.7)

and the discontinuity propagates with the speed $\sigma$.

There are two different waves associated with the fields
$\lambda_{k}=\frac{v_{0}}{u_{0}},k=2,3$, with the corresponding linearly
independent right eigenvectors $\textbf{r}_{k},k=2,3$, in (2.6):

Vortex sheets:

$C_{2}(U_{0}):\hskip 8.53581pt\sigma=\frac{v}{u}=\frac{v_{0}}{u_{0}},\hskip
11.38109ptp=p_{0},\hskip 8.53581ptS=S_{0},\hskip 8.53581ptu^{2}+v^{2}\neq
u_{0}^{2}+v^{2}_{0},$ (2.8)

Entropy waves:

$C_{3}(U_{0}):\hskip 8.53581pt\sigma=\frac{v}{u}=\frac{v_{0}}{u_{0}},\hskip
11.38109ptp=p_{0},\hskip 8.53581pt(u,v)=(u_{0},v_{0}),\hskip 8.53581ptS\neq
S_{0}.$ (2.9)

Albeit the two contact discontinuities, the vortex sheet and the entropy wave,
above match as a single discontinuity in the physical $(x,y)$–plane, two
independent parameters are needed to describe them in the phase space
$U=(u,v,p,\rho)$ since there are two linearly independent eigenvectors
corresponding to the repeated eigenvalues
$\lambda_{2}=\lambda_{3}=\frac{v}{u}$ of the linearly degenerate
characteristics fields.

The nonlinear waves associated with $\lambda_{d},d=1,4$, are shock waves and
rarefaction waves. The shock waves have their speeds of propagation given by

$\sigma=\sigma_{d}\mathrel{\mathop{:}}=\frac{u_{0}v_{0}+(-1)^{d}\bar{c}_{0}\sqrt{u_{0}^{2}+v_{0}^{2}-\bar{c}_{0}^{2}}}{u^{2}-\bar{c}_{0}^{2}},\qquad
d=1,4,$

where $\bar{c}_{0}^{2}=\frac{c^{2}_{0}}{b_{0}}\frac{\rho}{\rho_{0}}$ and
$b_{0}=\frac{\gamma+1}{2}-\frac{\gamma-1}{2}\frac{\rho}{\rho_{0}}$.
Substituting $\sigma_{d},$ $d=1,4$, into (2.7), the $d$-Hugoniot curve
$S_{d}(U_{0})$ through the state $U_{0}$ is

$S_{d}(U_{0}):\hskip 8.53581pt[p]=\frac{c^{2}_{0}}{b_{0}}[\rho],\hskip
8.53581pt[u]=-\sigma_{d}[v],\hskip
8.53581pt\rho_{0}(\sigma_{d}u_{0}-v_{0})[v]=[p],\qquad d=1,4.$

Written as $S_{d}^{+}(U_{0})$, $d=1,4$, the half curves of $S_{d}(U_{0})$ for
$\rho>\rho_{0}$ in the phase space are said to be the shock curves on which
any state forms a shock with the below state $U_{0}$ in the $(x,y)$–plane
respecting the entropy condition (1.9). Furthermore, for each $d=1$ or $d=4$,
the curves $S^{+}_{d}(U_{0})$ and $R^{-}_{d}(U_{0})$ at the state $U_{0}$ have
the same curvature.

If $U$ is a piecewise smooth solution (see also [11]), then any of the
following conditions below is equivalent to the entropy inequality (1.9) in
Definition 1.1 for a shock wave:

  * (i)

The physical entropy condition: The density increases across the shock in the
flow direction,

$\rho_{\text{back}}<\rho_{\text{front}}.$ (2.10)

  * (ii)

The Lax entropy condition: On the $d$th-shock, the shock speed $\sigma_{d}$
satisfies

$\displaystyle\lambda_{d}(\text{back})<\sigma_{d}<\lambda_{d}(\text{front})\hskip
8.53581pt\mbox{for}\,\,\,d=1,4,$ (2.11)
$\displaystyle\sigma_{1}<\lambda_{2,3}(\text{back}),\hskip
14.22636pt\lambda_{2,3}(\text{front})<\sigma_{4}.$ (2.12)

The rarefaction wave curves $R_{l}^{-}(U_{0})$ through the state $U_{0}$ in
the state space are given by

$R^{-}_{d}:\hskip 8.53581ptdp=c^{2}d\rho,\hskip
8.53581ptdu=-\lambda_{d}dv,\hskip 8.53581pt\rho(\lambda_{d}u-v)dv=dp\hskip
8.53581pt\text{ for }\rho<\rho_{0},\qquad d=1,4.$ (2.13)

We next discuss several essential properties of the nonlinear waves and
related wave interaction estimates in Lemmas 2.1–2.7 below. These facts will
be used in the later sections. We also refer the reader to Chen-Zhang-Zhu [11]
for further details.

_2.1. Riemann Problems and Riemann Solutions_

We focus on the related Riemann problems and their solutions in this section,
which serve as the building blocks for the front tracking algorithm for the
initial-boundary value problem (2.1) and (1.7)–(1.8).

_Riemann problem of lateral-type._ We note that the straight-sided wall
problem is the case when problem (2.1) and (1.7)–(1.8) is considered with the
boundary $g\equiv 0$. It can be seen that, when the angle between the
straight-sided wall and the flow direction of the incoming flow is zero,
problem (2.1) and (1.7)–(1.8) has an entropy solution made up of two constants
states $U_{-}=(u_{-},0,p_{-},\rho_{-})$ and $U_{+}=(u_{+},0,p_{+},\rho_{+})$,
satisfying $u_{\pm}>c_{\pm}>0$ in the subdomains $\Omega_{+}$ and $\Omega_{-}$
of $\Omega$ separated by a straight vortex sheet/entropy wave. These are
precisely the states $U_{-}$ and $U_{+}$ below and above the large vortex
sheet/entropy wave. The principal aim of this paper is to establish the
$L^{1}$–well-posedness for problem (2.1) and (1.7)–(1.8) for the solutions
near the background solution containing a strong vortex sheet/entropy wave
$\\{U_{-},U_{+}\\}$ with $g\equiv 0$.

It has been observed in [12] that, if the angle between the flow direction of
the front state and the wall at a boundary vertex is smaller than $\pi$ and
larger than the extreme angle determined by the incoming flow state and
$\gamma\geq 1$, then a unique $4$-shock is generated, separating the front-
state from the supersonic back-state. If the angle between the flow direction
of the front-state and the wall at a boundary vertex is larger than $\pi$ and
less than the extreme angle, then a $4$-rarefaction wave is produced,
emanating from the vertex. These waves are easily seen through the shock polar
analysis (cf. [11, 12]). This signifies that, when the angle between the flow
direction of the front-state and the wall at a boundary vertex is close to
$\pi$, the lateral Riemann problem can be uniquely solved. For further
details, see Lemma 2.3 and [11]. For an indepth discussion, we also refer to
Courant–Friedrichs [12].

_Riemann problem involving only weak waves._ Consider the subsequent initial
value problem with piecewise constant initial data:

$\begin{cases}W(U)_{x}+H(U)_{y}=0,\\\\[5.69054pt]
U|_{x=x_{0}}=\underline{U}=\begin{cases}U_{a},\hskip 5.69054pty>y_{0},\\\
U_{b},\hskip 5.69054pty<y_{0},\end{cases}\end{cases}$ (2.14)

with the constant states $U_{a}$ and $U_{b}$ denoting the above state and
below state with respect to the line $y=y_{0}$, respectively. Then there is
$\varepsilon>0$ so that, for any states $U_{b},U_{a}$ in the neighborhood
$\textit{O}_{\varepsilon}(U_{+})$ of $U_{+}$, or $U_{b},U_{a}$ in the
neighborhood $\textit{O}_{\varepsilon}(U_{-})$ of $U_{-}$, the initial value
problem (2.14) has a unique admissible solution consisting of four waves,
consisting of shocks, rarefaction waves, vortex sheets and/or entropy waves.

_Riemann problem involving the strong vortex sheets/entropy waves._ From now
on, the notation
$\\{U_{b},U_{a}\\}=\left(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\right)$
will be used to write
$U_{a}={\Phi}\left(\alpha_{4},\alpha_{3},\alpha_{2},\alpha_{1};U_{b}\right)$
as the solution of the Riemann problem, where ${\Phi}\in C^{2}$, and
$\alpha_{j}$ is the strength of the $j$-wave. For any wave with $U_{b}\in
O_{\varepsilon}(U_{-})$ and $U_{a}\in O_{\varepsilon}(U_{+})$, we also use
$\\{U_{b},U_{a}\\}=\left(0,\sigma_{2},\sigma_{3},0\right)$ to denote the
strong vortex sheet/entropy wave that connects $U_{b}$ and $U_{a}$ with
strength $\left(\sigma_{2},\sigma_{3}\right)$. That is,

$U_{m}=\Phi(\sigma_{2};U_{b})\mathrel{\mathop{:}}=\left(u_{b}e^{\sigma_{2}},v_{b}e^{\sigma_{2}},p_{b},\rho_{b}\right),\quad
U_{a}=\Phi(\sigma_{3};U_{m})\mathrel{\mathop{:}}=\left(u_{m},v_{m},p_{m},\rho_{m}e^{\sigma_{3}}\right).$

Particularly, we observe that

$U_{+}=(u_{+},0,p_{+},\rho_{+})=(u_{-}e^{\sigma_{20}},0,p_{-},\rho_{-}e^{\sigma_{30}}).$

We write
$G\left(\sigma_{3},\sigma_{2};U_{b}\right)=\Phi_{3}(\sigma_{3};\Phi_{2}(\sigma_{2};U_{b}))$
for any $U_{b}\in O_{\varepsilon}(U_{-})$. Then we have

Lemma 2.1. _The vector function G( $\sigma_{3},\sigma_{2};U_{b}$) satisfies_

$G_{\sigma_{2}}\left(\sigma_{3},\sigma_{2};U_{b}\right)=\left(u_{b}e^{\sigma_{2}},v_{b}e^{\sigma_{2}},0,0\right),\hskip
11.38109ptG_{\sigma_{3}}\left(\sigma_{3},\sigma_{2};U_{b}\right)=(0,0,0,\rho_{b}e^{\sigma_{3}}),$
(2.15)

_and_

$\nabla_{U}G(\sigma_{3},\sigma_{2};U_{b})={\rm
diag}(e^{\sigma_{2}},e^{\sigma_{2}},1,e^{\sigma_{3}}).$ (2.16)

_Furthermore, for the plane vortex sheet and entropy wave with the lower state
$U_{-}=(u_{-},0,p_{-},\rho_{-})$, upper state
$U_{+}=(u_{+},0,p_{+},\rho_{+})$, and strength $(\sigma_{20},\sigma_{30})$_,

${\rm
det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))>0.$
(2.17)

These can be easily obtained from direct calculations and are thus omitted.

The properties in (2.15)–(2.17) above play a fundamental role in achieving the
necessary estimates on the strengths of reflected weak waves in the
interaction between the strong vortex sheet/entropy wave and weak waves (see
the proofs for Lemmas 2.4–2.7).

_2.2. Wave Interactions and Reflection Estimates._ In the following, several
essential estimates are provided on wave interactions and reflections. For
their proofs and all the related details, we refer to [11].

Weak wave interactions estimates. For the weak wave interaction away from both
the strong vortex sheet/entropy wave and the wall boundary in the regions
$\Omega_{+}$ or $\Omega_{-}$, we have the following estimate:

Lemma 2.2. _Assume that_

$U_{b},U_{m},U_{a}\in O_{\varepsilon}(U_{+}),\quad\mbox{or}\quad
U_{b},U_{m},U_{a}\in O_{\varepsilon}(U_{-}),$

_are three states with
$\\{U_{b},U_{m}\\}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})$,
$\\{U_{m},U_{a}\\}=(\beta_{1},\beta_{2},\beta_{3},\beta_{4})$. Then
$\\{U_{b},U_{a}\\}=(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$ with_

$\gamma_{i}=\alpha_{i}+\beta_{i}+O(1)\Delta(\alpha,\beta),$ (2.18)

_where
$\Delta(\alpha,\beta)=(|\alpha_{4}|+|\alpha_{3}|+|\alpha_{2}|)|\beta_{1}|+|\alpha_{4}|(|\beta_{2}|+|\beta_{3}|)+\sum_{j=1,4}\Delta_{j}(\alpha,\beta)$
and_

$\Delta_{j}(\alpha,\beta)=\begin{cases}0,\hskip 8.53581pt&\alpha_{j}\geq
0,\beta_{j}\geq 0,\\\ |\alpha_{j}||\beta_{j}|,\hskip
5.69054pt&otherwise.\end{cases}$ (2.19)

Estimates on the boundary perturbation of weak waves and the reflection of
weak waves on the boundary. We write $\\{C_{l}(a_{l},b_{l})\\}^{\infty}_{l=0}$
for the points $\\{(a_{l},b_{l})\\}^{\infty}_{l=0}$ in the $(x,y)$–plane with
$0<a_{l}<a_{l+1}$. Define

$\displaystyle\left\\{\begin{array}[]{ll}\theta_{{l},{l+1}}=\text{arctan}\left(\frac{b_{l+1}-b_{l}}{a_{l+1}-a_{l}}\right),\hskip
5.69054pt\theta_{l}=\theta_{{l},{l+1}}-\theta_{{l-1},{l}},\hskip
5.69054pt\theta_{-1,0}=0,\\\\[5.69054pt]
\Omega_{l+1}=\\{(x,y):x\in\left[a_{l},a_{l+1}\right],y>b_{l}+(x-a_{l})\text{tan}(\theta_{{l},{l+1}})\\},\\\\[5.69054pt]
\Gamma_{l+1}=\\{(x,y):x\in\left(a_{l},a_{l+1}\right),y=b_{l}+(x-a_{l})\text{tan}(\theta_{{l},{l+1}})\\},\end{array}\right.$
(2.23)

and the outer normal vector to $\Gamma_{l}$:

$\textbf{n}_{l+1}=\frac{(b_{l+1}-b_{l},a_{l}-a_{l+1})}{\sqrt{(b_{l+1}-b_{l})^{2}+(a_{l+1}-a_{l})^{2}}}=(\text{sin}(\theta_{l},\theta_{l+1}),-\text{cos}(\theta_{l},\theta_{l+1})).$
(2.24)

With the constant state $\underline{U}$, consider the following initial-
boundary value problem:

$\begin{cases}(2.1)\hskip 62.59596pt&\text{in}\hskip 2.84526pt\Omega_{l+1},\\\
U|_{x=a_{l}}=\underline{U},\\\ (u,v)\cdot\textbf{n}_{l+1}=0\hskip
8.53581pt&\text{on}\hskip 2.84526pt\Gamma_{l+1}.\end{cases}$ (2.25)

Lemma 2.3. _Suppose
$\left\\{U_{m},U_{a}\right\\}=(\beta_{1},\beta_{2},\beta_{3},0)$ and
$\left\\{U_{l},U_{m}\right\\}=(0,0,0,\alpha_{4})$ with_

$(u_{l},v_{l})\cdot\textbf{n}_{l}=0.$

_Then there exists a unique solution $U_{l+1}$ of problem (2.25) such that
$\left\\{U_{l+1},U_{a}\right\\}=(0,0,0,\delta_{4})$ and
$U_{l+1}\cdot(\textbf{n}_{l+1},0,0)=0$. Moreover,_

$\delta_{4}=\alpha_{4}+K_{b1}\beta_{1}+K_{b2}\beta_{2}+K_{b3}\beta_{3}+K_{b0}\theta_{l},$
(2.26)

_where $K_{b1}$, $K_{b2}$, $K_{b3}$, and $K_{b0}$ are $C^{2}$-functions of
$\beta_{3}$, $\beta_{2}$, $\beta_{1}$, $\alpha_{4}$, $\theta_{l+1}$, and
$U_{a}$ satisfying_

$K_{b1}|_{\left\\{\theta_{l}=\alpha_{4}=\beta_{1}=\beta_{2}=\beta_{3}=0,U_{a}=U_{-}\right\\}}=1,\hskip
28.45274ptK_{bi}|_{\left\\{\theta_{l}=\alpha_{4}=\beta_{1}=\beta_{2}=\beta_{3}=0,U_{a}=U_{-}\right\\}}=0,\,\,i=2,3,$
(2.27)

_and $K_{b0}$ is bounded. In particular, $K_{b0}<0$ at the origin._

This lemma has two purposes. The first is to estimate the weak waves generated
by the vertices on the Lipschitz wall boundary. This boundedness will be used
to control the boundary perturbation; see (3.2) in the construction of the
wave interaction potential $\mathcal{Q}(x)$. The second is to estimate the
strength of the reflected wave $\delta_{4}$ with respect to the incident wave
$\alpha_{1}$. Property (2.27) of the coefficients will play an important role
to control the reflected waves.

Estimates on the interaction between the strong vortex sheet/entropy wave and
weak waves from below. Estimate (2.28) below plays a key role in ensuring the
$L^{1}$–stability of entropy solutions, especially for the existence of the
constants $w^{b}_{1}$ and $w^{b}_{4}$ in Lemma 5.1 (see below). This estimate
also ensures the existence of $K^{\ast}\in$ ($K_{11}$, 1) in the construction
of the wave interaction potential $\mathcal{Q}(x)$ in (3.2).

Lemma 2.4. _Let $U_{b},U_{m}\in O_{\varepsilon}(U_{-})$ and $U_{a}\in
O_{\varepsilon}(U_{+})$ with_

$\\{U_{b},U_{m}\\}=(0,\alpha_{2},\alpha_{3},\alpha_{4}),\hskip
19.91692pt\\{U_{m},U_{a}\\}=(\beta_{1},\sigma_{2},\sigma_{3},0).$

_Then there exists a unique
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ such that
the Riemann problem (2.14) admits an admissible solution that consists of a
weak $1-$wave of strength $\delta_{1}$, a strong vortex sheet of strength
$\sigma_{2}$, a strong entropy wave of strength $\sigma_{3}$, and a weak
$4-$wave of strength $\delta_{4}$:_

$\\{U_{b},U_{a}\\}=(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4}).$

_Moreover,_

$\displaystyle\delta_{1}=\beta_{1}+K_{11}\alpha_{4}+O(1)\Delta^{\prime},\hskip
14.22636pt\delta_{4}=K_{14}\alpha_{4}+O(1)\Delta^{\prime},$
$\displaystyle\sigma_{2}^{\prime}=\sigma_{2}+\alpha_{2}+K_{12}\alpha_{4}+O(1)\Delta^{\prime},\hskip
14.22636pt\sigma_{3}^{\prime}=\sigma_{3}+\alpha_{3}+K_{13}\alpha_{4}+O(1)\Delta^{\prime},$
$\displaystyle|K_{11}|_{\left\\{\alpha_{4}=\alpha_{3}=\alpha_{2}=0,\sigma_{2}^{\prime}=\sigma_{20},\sigma_{3}^{\prime}=\sigma_{30}\right\\}}=\left|\frac{\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{4}(U_{-})}{\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-})}\right|<1,$
(2.28)

_where $\Delta^{\prime}=|\beta_{1}|(|\alpha_{2}|+|\alpha_{3}|)$, and
$\sum_{j=2}^{4}|K_{1j}|$ is bounded._

Lemma 2.5. _The coefficient
$|K_{14}|_{\left\\{\alpha_{4}=\alpha_{3}=\alpha_{2}=0,\sigma_{2}^{\prime}=\sigma_{20},\sigma_{3}^{\prime}=\sigma_{30}\right\\}}$
in the strength $\delta_{4}$ of a weak 4-wave in Lemma 2.4 remains bounded
away from zero._

Proof. By Lemma 2.4, we can find a unique solution
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ as a
$C^{2}$-function of $\alpha_{2},\alpha_{3},\alpha_{4},\beta_{1},$
$\sigma_{2},\sigma_{3}$, and $U_{b}$ to

$\Phi_{4}(\delta_{4};G(\sigma^{\prime}_{3},\sigma^{\prime}_{2};\Phi_{1}(\delta;U_{b})))=G(\sigma_{2},\sigma_{3};\Phi_{1}(\beta_{1},\Phi(\alpha_{4},\alpha_{3},\alpha_{2},0;U_{b}))).$
(2.29)

That is,

$\sigma^{\prime}_{i}=\sigma^{\prime}_{i}(\alpha_{2},\alpha_{3},\alpha_{4},\beta_{1},\sigma_{2},\sigma_{3}),\hskip
5.69054pti=2,3;\hskip
14.22636pt\delta_{j}=\delta_{j}(\alpha_{2},\alpha_{3},\alpha_{4},\beta_{1},\sigma_{2},\sigma_{3}),\hskip
5.69054ptj=1,4,$

where we have omitted $U_{b}$ for simplicity.

From [11], we know that

$K_{1j}=\int\limits_{0}^{1}\
\partial_{\alpha_{4}}\delta_{j}(\alpha_{2},\alpha_{3},\theta\alpha_{4},\beta_{1},\sigma_{2},\sigma_{3})\,d\theta,\hskip
14.22636ptj=1,4.$

Differentiate (2.29) with respect to $\alpha_{4}$, and let
$\beta_{1}=\alpha_{4}=\alpha_{3}=\alpha_{2}=0,\sigma_{2}=\sigma_{20}$, and
$\sigma_{3}=\sigma_{30}$. We obtain

$\displaystyle\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{4}(U_{-})$
$\displaystyle=$
$\displaystyle\partial_{\alpha_{4}}\delta_{4}\,\textbf{r}_{4}(U_{+})+\partial_{\alpha_{4}}\sigma^{\prime}_{3}\,G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-})$
$\displaystyle+\hskip
2.84526pt\partial_{\alpha_{4}}\sigma^{\prime}_{2}\,G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-})+\partial_{\alpha_{4}}\delta_{1}\,\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}).$

By Lemma 2.1, we have

$|\partial_{\alpha_{4}}\delta_{4}|$

$\displaystyle=$
$\displaystyle\left|\frac{\text{det}(\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{4}(U_{-}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\kappa_{1}(U_{-})\kappa_{4}(U_{-})\rho^{2}_{-}u^{2}_{-}e^{2\sigma_{20}+\sigma_{30}}(\lambda_{4}(U_{-})-\lambda_{1}(U_{-}))}{\kappa_{1}(U_{-})\kappa_{4}(U_{+})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}(\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{2\kappa_{4}(U_{-})e^{\sigma_{20}}\lambda_{4}(U_{-})}{\kappa_{4}(U_{+})(\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-}))}\right|>0.$

This completes the proof.

_Estimates on the interaction between the strong vortex sheet/entropy wave and
weak waves from above._ We have

Lemma 2.6. _Let $U_{b}\in O_{\varepsilon}(U_{-})$ and $U_{m},U_{a}\in
O_{\varepsilon}(U_{+})$ with_

$\\{U_{b},U_{m}\\}=(0,\sigma_{2},\sigma_{3},\alpha_{4}),\hskip
19.91692pt\\{U_{m},U_{a}\\}=(\beta_{1},\beta_{2},\beta_{3},0).$

_Then there exists a unique
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ such that
the Riemann problem (2.14) admits an admissible solution that consists of a
weak $1-$wave of strength $\delta_{1}$, a strong vortex sheet of strength
$\sigma_{2}$, a strong entropy wave of strength $\sigma_{3}$, and a weak
$4-$wave of strength $\delta_{4}$:_

$\\{U_{b},U_{a}\\}=(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4}).$

_Moreover,_

$\delta_{1}=K_{21}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},\hskip
14.22636pt\sigma_{2}^{\prime}=\sigma_{2}+\beta_{2}+K_{22}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},$
$\sigma_{3}^{\prime}=\sigma_{3}+\beta_{3}+K_{23}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},\hskip
14.22636pt\delta_{4}=\alpha_{4}+K_{24}\beta_{1}+O(1)\Delta^{{}^{\prime\prime}},$

_where $\sum_{j=1}^{4}|K_{2j}|$ is bounded and
$\Delta^{{}^{\prime\prime}}=|\alpha_{4}|(|\beta_{2}|+|\beta_{3}|).$_

The constant $K_{21}$ here is used in the definition of weighted strength
$b_{\alpha}$ of weak waves in (3.1).

Lemma 2.7. _The coefficient
$|K_{21}|_{\left\\{\beta_{3}=\beta_{2}=\beta_{1}=0,\sigma_{2}^{\prime}=\sigma_{20},\sigma_{3}^{\prime}=\sigma_{30}\right\\}}$
in the strength $\delta_{1}$ of a weak 1-wave in Lemma 2.6 remains bounded
away from zero, while the reflection coefficient $|K_{24}|<1$._

Proof. By Lemma 2.6, we can find a unique solution
$(\delta_{1},\sigma^{\prime}_{2},\sigma^{\prime}_{3},\delta_{4})$ as a
$C^{2}$-function of $\alpha_{2},\alpha_{3},\alpha_{4}$, $\beta_{1}$,
$\sigma_{2}$, $\sigma_{3}$, and $U_{b}$ to

$\Phi_{4}(\delta_{4};G(\sigma^{\prime}_{3},\sigma^{\prime}_{2};\Phi_{1}(\delta;U_{b})))=\Phi(0,\beta_{3},\beta_{2},\beta_{1};\Phi_{4}(\alpha_{4};G(\sigma_{3},\sigma_{2};U_{b}))).$
(2.30)

That is,

$\sigma^{\prime}_{i}=\sigma^{\prime}_{i}(\beta_{1},\beta_{2},\beta_{3},\alpha_{4},\sigma_{2},\sigma_{3}),\hskip
5.69054pti=2,3;\hskip
14.22636pt\delta_{j}=\delta_{j}(\beta_{1},\beta_{2},\beta_{3},\alpha_{4},\sigma_{2},\sigma_{3}),\hskip
5.69054ptj=1,4,$

where we have omitted $U_{b}$ for simplicity.

From [11], we know that

$K_{2j}=\int\limits_{0}^{1}\
\partial_{\beta_{1}}\partial_{j}(\theta\beta_{1},\beta_{2},\beta_{3},\alpha_{4},\sigma_{2},\sigma_{3})\,d\theta,\hskip
8.53581ptj=1,4.$

Differentiate (2.30) with respect to $\beta_{1}$ and let
$\alpha_{4}=\beta_{1}=\beta_{2}=\beta_{3}=0,\sigma_{2}=\sigma_{20}$, and
$\sigma_{3}=\sigma_{30}$. We obtain

$\displaystyle\textbf{r}_{1}(U_{+})$ $\displaystyle=$
$\displaystyle\partial_{\beta_{1}}\delta_{4}\,\textbf{r}_{4}(U_{+})+\partial_{\beta_{1}}\sigma^{\prime}_{3}\,G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-})$
$\displaystyle+\hskip
2.84526pt\partial_{\beta_{1}}\sigma^{\prime}_{2}\,G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-})+\partial_{\beta_{1}}\delta_{1}\,\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}).$

By Lemma 2.1, we have

$\displaystyle|\partial_{\beta_{1}}\delta_{1}|$ $\displaystyle=$
$\displaystyle\left|\frac{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\textbf{r}_{1}(U_{+}))}{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\kappa_{4}(U_{+})\kappa_{1}(U_{+})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}\lambda_{4}(U_{+})e^{\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{+})e^{\sigma_{20}+\sigma_{30}}\big{)})}{\kappa_{4}(U_{+})\kappa_{1}(U_{-})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{-})\big{)}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{2\kappa_{1}(U_{+})\lambda_{4}(U_{+})e^{\sigma_{20}+\sigma_{30}}}{\kappa_{1}(U_{-})(\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-}))}\right|>0.$

However, for the reflection coefficient $|K_{24}|$, we have

$\displaystyle|\partial_{\beta_{1}}\delta_{4}|$ $\displaystyle=$
$\displaystyle\left|\frac{\text{det}(\textbf{r}_{1}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}{\text{det}(\textbf{r}_{4}(U_{+}),G_{\sigma_{3}}(\sigma_{30},\sigma_{20};U_{-}),G_{\sigma_{2}}(\sigma_{30},\sigma_{20};U_{-}),\nabla_{U}G(\sigma_{30},\sigma_{20};U_{-})\cdot\textbf{r}_{1}(U_{-}))}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{\kappa_{1}(U_{-})\kappa_{1}(U_{+})\rho_{-}u_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}-\rho_{-}u_{-}\lambda_{1}(U_{-})+e^{\sigma_{20}}\rho_{+}u_{+}\lambda_{1}(U_{+})\big{)}}{\kappa_{4}(U_{+})\kappa_{1}(U_{-})\rho^{2}_{-}u^{2}_{-}e^{\sigma_{20}+\sigma_{30}}\big{(}\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}-\lambda_{1}(U_{-})\big{)}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{-\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-})}{\lambda_{4}(U_{+})e^{2\sigma_{20}+\sigma_{30}}+\lambda_{4}(U_{-})}\right|<1,$

where $|K_{24}|$ is not necessarily bounded away from zero, but is less than
one.

##  3\. The Wave-Front Tracking Algorithm and Global Existence of Weak
Solutions

We start off here with a brief description of the wave-front tracking method
to be employed throughout in Sections 4–7 and then establish the existence of
entropy solutions when the perturbation of the incoming flow has small total
variation at $x=0$.

The main scheme in the method of wave-front tracking is to construct
approximate solutions within a class of piecewise constant functions. At the
beginning, we approximate the initial data by a piecewise constant function.
Then we solve the resulting Riemann problems exactly with the exception of the
rarefaction waves, which are replaced by rarefaction fans with many small
wave-fronts of equal strengths. The outgoing fronts are continued up to the
first time when two waves collide and a new Riemann problem is solved. In this
process, one has to modify the algorithm and introduce a simplified Riemann
solver in order to keep the number of wave-fronts finite for all $x\geq 0$ in
the flow direction. We refer the reader to Bressan [4, 6] and Baiti-Jenssen
[2] for related references.

_3.1. The Riemann Solvers_. As seen in Section 2, the solution to the Riemann
problem $\\{U_{b},U_{a}\\}$ is a self-similar solution given by at most five
states separated by shocks, vortex sheets, entropy waves, or rarefaction
waves. To connect the state $U_{a}$ to $U_{b}$, there exist $C^{2}$–curves
$\eta\rightarrow\varphi(\eta)(U)$ with arc length parametrization such that

$U_{b}=\varphi(\eta)(U_{a}):=\Upsilon_{4}(\eta_{4})\circ\cdots\circ\Upsilon_{1}(\eta_{1})(U_{a})$

for some $\eta=(\eta_{1},\ldots,\eta_{4})$, and
$U_{j}=\Upsilon_{j}(\eta_{j})\circ\cdots\circ\Upsilon_{1}(\eta_{1})(U_{a})$,
$j=1\ldots 3$.

Next, we discuss the construction of front tracking approximations for our
initial-boundary value problem. Let $\vartheta$ denote the initial
approximation parameter. For given initial data $\overline{U}$ and with
$\vartheta>0$, consider $\overline{U}^{\vartheta}$ a sequence of piecewise
constant functions approximating $\overline{U}$ in the $L^{1}$–norm, and the
wall boundary is also approximated as described in Section 2. Set
$\mathcal{Z}_{\vartheta}$ to be the total number of jumps in the initial data
$\overline{U}^{\vartheta}$ and the tangential angle function of the wall
boundary. Let $\delta_{\vartheta}>0$ be a parameter so that a rarafaction wave
is replaced by a step function whose “steps” are no further apart than
$\delta_{\vartheta}$. The discontinuity between two steps is set to propagate
with a speed equal to the Rankine-Hugoniot speed of the jump connecting the
states corresponding to the two steps. At any time, the simplified Riemann
solver (defined below) is employed, the constant $\hat{\lambda}$ denotes the
speed of the generated non-physical wave, which is strictly greater than all
the wave speeds of system (2.1). Note that the strength of the non-physical
wave is the error generated when the simplified Riemann solver is applied.

Accurate Riemann solver. The accurate Riemann solver (ARS) is the exact
solution to the Riemann problem, with the condition that every rarefaction
wave $\\{w,R_{d}(w)(\alpha)\\}$, $d=1,4$, is divided into equal parts and
replaced by a piecewise constant rarefaction fan of several new wave-fronts of
equal strength.

Simplified Riemann solver. When only weak waves are involved, the simplified
Riemann solver (SRS) here is the same as the one described in [2, 6]. That is,
all new waves are put together in a single non-physical front, travelling
faster than all characteristic speeds. In the case of a weak wave interacting
with the strong vortex sheet/entropy wave, the purpose of (SRS) is to ignore
the strength of the weak wave, while preserving the strength of the strong
vortex sheet/entropy wave, and to place the error in the non-physical wave in
the following manner:

Case 1 : A weak wave $\\{U_{-},U_{1}\\}$ collides with the strong vortex
sheet/entropy wave $\\{U_{1},U_{+}\\}$ from below. The Riemann problem
$\\{U_{-},U_{+}\\}$ is solved as follows:

$\left\\{\begin{array}[]{ll}U_{-}&\quad\mbox{for
$\frac{y}{x}<\chi(U_{1},U_{+})$},\\\ U_{2}&\quad\mbox{for
$\chi(U_{1},U_{+})<\frac{y}{x}<\hat{\lambda}$},\\\ U_{+}&\quad\mbox{for
$\frac{y}{x}>\hat{\lambda}$},\end{array}\right.$

with $\chi(U_{1},U_{+})$ as the speed of the strong vortex sheet/entropy wave,
and the state $U_{2}$ is solved in a way that $\\{U_{-},U_{2}\\}$ is the
strong vortex sheet/entropy wave starting from $U_{-}$ and
$\chi(U_{1},U_{+})=\chi(U_{-},U_{2})$. Hence, we find that (SRS) keeps the
same strength of the strong vortex sheet/entropy wave, and the error appears
in the non-physical fronts.

Case 2: A weak wave $\\{U_{2},U_{+}\\}$ collides with the strong vortex
sheet/entropy wave $\\{U_{-},U_{2}\\}$ from above. The Riemann problem
$\\{U_{-},U_{+}\\}$ is solved as follows:

$\left\\{\begin{array}[]{ll}U_{-}&\quad\mbox{for
$\frac{y}{x}<\chi(U_{-},U_{2})$},\\\ U_{2}&\quad\mbox{for
$\chi(U_{-},U_{2})<\frac{y}{x}<\hat{\lambda}$},\\\ U_{+}&\quad\mbox{for
$\frac{y}{x}>\hat{\lambda}$},\end{array}\right.$

with $\chi(U_{-},U_{2})$ denoting the speed of the strong vortex sheet/entropy
wave.

_3.2. Construction of Wave Front Tracking Approximations_

Given $\vartheta$, the corresponding front tracking approximate solution
$U^{\vartheta}(x,y)$ is built up as follows. At $x=0$, all the Riemann
problems in $\overline{U}^{\vartheta}$ are solved by using the accurate
Riemann solver. Furthermore, one can change the speed of one of the incoming
fronts so that, at any time $x>0$, there is at most one collision involving
only two incoming fronts. Of course, this adjustment of speed can be chosen
arbitrarily small. Let $\omega_{\vartheta}$ be a fixed small parameter with
$\omega_{\vartheta}\rightarrow 0$, as $\vartheta\rightarrow 0$, which will be
determined later. For convenience, the index $j$ in $\alpha_{j}$ will be
dropped henceforward, and we will write $\alpha_{j}$ as $\alpha$ when there is
no ambiguity involved; the same applies for $\beta$; and we will moreover
employ the same notation $\alpha$ as a wave and its strength as before.

_Case 1: Two weak waves with strengths $\alpha$ and $\beta$ interact at some
$x>0$_. The Riemann problem produced by this collision is solved in the
following way:

  * •

If $|\alpha\beta|>\omega_{\vartheta}$ and the two waves are physical, then the
accurate Riemann solver is employed.

  * •

If $|\alpha\beta|<\omega_{\vartheta}$ and the two waves are physical, or there
is a non-physical wave, then the simplified Riemann solver is employed.

_Case 2: A weak wave $\alpha$ interacts with the strong vortex sheet/entropy
wave and one weak wave at some $x>0$_. The Riemann problem produced by this
collision is solved in the following way:

  * •

If $|\alpha|>\omega_{\vartheta}$ and the weak wave is physical, then the
accurate Riemann solver is applied.

  * •

If $|\alpha|<\omega_{\vartheta}$ and the weak wave is physical, or this wave
is non-physical, then the simplified Riemann solver is applied.

_Case 3: The flow perturbation due to the Lipschitz wall boundary_.

  * •

When the change of the angle of the boundary is larger than
$\omega_{\vartheta}$ and the weak wave is physical, then the accurate Riemann
solver is employed to solve the lateral Riemann problem.

  * •

If the change of the angle of the boundary is less than $\omega_{\vartheta}$,
then this perturbation is ignored.

_Case 4: The physical wave collides with the boundary_. The accurate Riemann
solver is employed to solve the lateral Riemann problem.

_Case 5: The non-physical wave collides with the boundary_. We can allow these
waves to cross the boundary.

_3.3. Glimm’s Functional and Wave Interaction Potential_

The goal in this subsection is to construct the suitable Glimm-type functional
and the associated wave interaction potential $\mathcal{Q}$ for our initial-
boundary value problem. This involves a careful incorporation of the
additional nonlinear waves generated from the wall boundary vertices.

Definition 3.1 (_Approaching waves_). (i) Two weak fronts $\alpha$ and
$\beta$, located at points $y_{\alpha}<y_{\beta}$ and of the characteristic
families $j_{\alpha}$, $j_{\beta}$ $\in$ $\\{1,\ldots,4\\}$, respectively, are
said to be approaching each other if the following two conditions are
concurrently satisfied:

  * •

$y_{\alpha}$ and $y_{\beta}$ are both in one of the two intervals into which
$\mathbb{R}$ is partitioned by the location of the strong vortex sheet/entropy
wave. That is, both waves are either in $\Omega_{-}$ or $\Omega_{+}$;

  * •

Either $j_{\alpha}>j_{\beta}$ or else $j_{\alpha}$ = $j_{\beta}$ and at least
one of them is a genuinely nonlinear shock.

In this case, we write $(\alpha,\beta)$ $\in$ $\mathcal{A}$.

(ii) We say that a weak wave $\alpha$ of the characteristic family
$j_{\alpha}$ is approaching the strong vortex sheet/entropy wave if either
$\alpha\in\Omega_{-}$ and $j_{\alpha}=4$, or $\alpha\in\Omega_{+}$ and
$j_{\alpha}=1$. We then write $\alpha\in\mathcal{A}_{v/e}$.

(iii) We say that a weak wave $\alpha$ of the characteristic family
$j_{\alpha}$ is approaching the boundary if $\alpha\in\Omega_{-}$ and
$j_{\alpha}=1$. We then write $\alpha\in\mathcal{A}_{b}$.

Define the total (weighted) strength of weak waves in $U^{\vartheta}(x,\cdot)$
as

$\mathcal{V}(x)=\sum_{\alpha}|b_{\alpha}|.$

Here, for a weak wave $\alpha$ of the $j$-family, its weighted strength is
defined as

$b_{\alpha}=\begin{cases}k_{+}\alpha&\quad\text{if $\alpha\in\Omega_{+}$ and
$j_{\alpha}=1$},\\\ \alpha&\quad\text{if $\alpha\in\Omega_{-}$},\end{cases}$
(3.1)

where $k_{+}=\frac{{2}K_{21}}{K^{*}}$ and the coefficient $K_{21}$ as given in
Lemma 2.6.

Next, the wave interaction potential $\mathcal{Q}(x)$ is defined as

$\displaystyle\mathcal{Q}(x)$ $\displaystyle=$ $\displaystyle
C^{\ast}\sum_{(\alpha,\beta)\in\mathcal{A}}|b_{\alpha}b_{\beta}|+K^{\ast}\sum_{\alpha\in\mathcal{A}_{v/e}}|b_{\alpha}|+\sum_{\beta\in\mathcal{A}_{b}}|b_{\beta}|+\widetilde{K_{b0}}\sum_{a_{l}>x}|\omega_{l}|$
(3.2) $\displaystyle=$
$\displaystyle\mathcal{Q}_{\mathcal{A}}+\mathcal{Q}_{v/e}+\mathcal{Q}_{b}+\mathcal{Q}_{\Theta}.$

Here the constants $K^{\ast}\in$ ($K_{11}$, 1) and
$\widetilde{K_{b0}}>K_{b0}$, while $C^{\ast}$ is a constant to be specified
later. To control the total variation of the new waves produced by the
boundary vertices, $\mathcal{Q}_{\Theta}$ in our wave interaction potential
$\mathcal{Q}(x)$ is an added term, compared to that for the Cauchy problem.

_The Glimm-type functional $\mathcal{G}$ is defined as follows_

$\mathcal{G}(x)=\mathcal{V}(x)+\kappa\mathcal{Q}(x)+|U^{\diamond}(x)-U^{+}_{0}|+|U_{\diamond}(x)-U^{-}_{0}|,$
(3.3)

_where the states $U_{\diamond}(x)$ and $U^{\diamond}(x)$ are the below state
and the above state of the strong vortex sheet/entropy wave respectively at
“time” x, $U^{-}_{0}$ and $U^{+}_{0}$ are the below and above state of the
strong vortex sheet/entropy wave respectively at $x=0$, and $\kappa$ is a
large positive constant to be determined later._

We remark that $\mathcal{V}$, $\mathcal{Q}$, and $\mathcal{G}$ remain
unchanged between any pair of subsequent interaction times. However, we will
demonstrate that, across an interaction “time” $x$, both $\mathcal{Q}$ and
$\mathcal{G}$ decrease.

Proposition 3.1. _Assume that ${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm
TV}(g^{\prime}\left(\cdot\right))$ is sufficiently small. Then
V$\left(x\right)$ will remain sufficiently small for all $x>0$. Moreover, the
quantity ${\rm TV}(U^{\vartheta}\left(x,\cdot\right))$ has a uniform bound for
any $\vartheta>0$_.

Proof. With the Glimm-type functional $\mathcal{G}$, consider

$\Delta\mathcal{G}(x)=\mathcal{G}(x^{+})-\mathcal{G}(x^{-}),$

where $x^{-}$ and $x^{+}$ denote the “times” before and after the interaction
“time” $x>0$, respectively.

_Case 1: Two weak waves $\alpha$ and $\beta$ collide_. Then the states
$U^{\diamond}\left(x\right)$ and $U_{\diamond}\left(x\right)$ do not alter
across this interaction “time” $x>0$. Hence, we have

$\displaystyle\Delta\mathcal{G}(x)$ $\displaystyle=$
$\displaystyle\mathcal{V}(x^{+})-\mathcal{V}(x^{-})+\kappa\left(\mathcal{Q}(x^{+})-\mathcal{Q}(x^{-})\right)$
$\displaystyle\leq$
$\displaystyle\mathcal{B}_{1}|b_{\alpha}b_{\beta}|+\kappa\left(-C^{\ast}|b_{\alpha}b_{\beta}|+C^{\ast}|b_{\alpha}b_{\beta}|V(x^{-})+\mathcal{B}_{0}|b_{\alpha}b_{\beta}|\right),$

where $\mathcal{B}_{0}$ and $\mathcal{B}_{1}$ are constants independent of
$\vartheta$.

_Case 2: A weak wave $\alpha$ of the 1-family interacts with the boundary_.

$\Delta\mathcal{G}(x)=K_{b1}\alpha-\alpha+\kappa\left(C^{*}K_{b1}V(x^{-})\alpha+K^{\ast}K_{b1}\alpha-\alpha\right),$

where $K^{\ast}K_{b1}<1$.

_Case 3: A new 4-wave $\alpha$ produced by the Lipschitz wall boundary_.

$\Delta\mathcal{G}(x)=K_{b0}\theta_{l}+\kappa\left(C^{\ast}K_{b0}\theta_{l}V(x^{-})+K^{*}K_{b0}\theta_{l}-\widetilde{K_{b0}}\theta_{l}\right),$

where $K_{b0}<\widetilde{K_{b0}}$ is large.

In the following two cases, the states $U_{\diamond}(x)$ and $U^{\diamond}(x)$
change across this interaction “time” $x>0$.

_Case 4: A weak wave $\alpha$ of the 4-family collides with the strong vortex
sheet/entropy wave from below_.

$\displaystyle\Delta\mathcal{G}(x)$ $\displaystyle=$
$\displaystyle\mathcal{V}(x^{+})-\mathcal{V}(x^{-})+|U^{\diamond}(x^{+})-U^{\diamond}(x^{-})|$
$\displaystyle+|U_{\diamond}(x^{+})-U_{\diamond}(x^{-})|+\kappa\left(\mathcal{Q}(x^{+})-\mathcal{Q}(x^{-})\right)$
$\displaystyle=$
$\displaystyle\sum^{4}_{j=1}K_{1j}\alpha-\alpha+\kappa\Big{(}C^{\ast}\big{(}K_{11}V(x^{-})\alpha+K_{14}V(x^{-})\alpha\big{)}-K^{\ast}\alpha+K_{11}\alpha\Big{)}.$

_Case 5: A weak wave $\alpha$ of the 1-family collides with the strong vortex
sheet/entropy wave from above_.

$\Delta\mathcal{G}(x)=\sum^{4}_{j=1}K_{2j}\alpha-
b_{\alpha}+\kappa\Big{(}C^{\ast}\big{(}K_{21}V(x^{-})\alpha+K_{24}V(x^{-})\alpha\big{)}-K^{\ast}b_{\alpha}+K_{21}\alpha\Big{)}.$

In the cases above, $K_{11}<K^{\ast}<1$, $b_{\alpha}>{2}K_{21}|\alpha|$ in
connection with the weight $k_{+}$, and the constant
$C^{\ast}>\mathcal{B}_{0}>0$ is large.

Next, we establish that the total (weighted) strength of waves in
$U^{\vartheta}(x,\cdot)$ remain sufficiently small for all $x>0$ if it is
sufficiently small at $x=0$. More precisely,

$\mathcal{V}(x)\ll 1\qquad\text{ for all }x>0.$

This can be proved as follows:

_(i) “Time” $x_{1}>0$ is the first interaction_. Given that
$\mathcal{V}\left(x^{-}_{1}\right)=\mathcal{V}(0)\leq{\rm
TV}(\widetilde{U_{0}}(\cdot))\ll 1$ and
$\sum_{\l=0}^{\infty}\theta_{l}\leq{\rm TV}(g^{\prime}(\cdot))\ll 1$ in Cases
1–5 above, we conclude that, for $\kappa$ sufficiently large and
$\omega_{\vartheta}$ small enough,

$\Delta\mathcal{G}(x_{1})\leq
0,\quad\text{i.e.,}\quad\mathcal{G}(x_{1}^{+})\leq\mathcal{G}(x_{1}^{-})=\mathcal{G}(0).$

Therefore,

$\displaystyle\mathcal{V}(x_{1}^{+})$ $\displaystyle\leq$
$\displaystyle\mathcal{G}(x_{1}^{+})\leq\mathcal{G}(0)\leq\mathcal{V}(0)+\kappa\mathcal{Q}(0)$
$\displaystyle=$
$\displaystyle\mathcal{V}(0)+\kappa\Big{(}C^{\ast}\mathcal{V}^{2}(0)+\mathcal{V}(0)+\widetilde{K_{b0}}\sum_{l=0}^{\infty}\theta_{l}\Big{)}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}\mathcal{V}(0)+\sum_{l=0}^{\infty}\theta_{l}\Big{)}\ll 1.$

_(ii) $\mathcal{V}(x_{m}^{-})\ll 1$ and
$\mathcal{G}(x_{m}^{+})\leq\mathcal{G}(x_{m}^{-})$ for any $m<n$_. Then, for
the next interaction “time” $x_{n}$, similar to Case 1, we also conclude

$\Delta\mathcal{G}(x_{n})\leq
0,\quad\text{i.e.,}\quad\mathcal{G}(x_{n}^{+})\leq\mathcal{G}(x_{n}^{-})=\mathcal{G}(x_{n-1}^{+}).$

Therefore, all together, we obtain

$\displaystyle\mathcal{V}(x_{n}^{+})+|U^{\diamond}(x_{n}^{+})-U_{0}^{+}|+|U_{\diamond}(x_{n}^{+})-U_{0}^{-}|\qquad\qquad$
$\displaystyle\qquad\leq\mathcal{G}(x_{n}^{+})\leq\mathcal{G}(x_{n}^{-})=\mathcal{G}(x_{n-1}^{+})\leq\ldots\leq\mathcal{G}(0)$
$\displaystyle\qquad=\mathcal{V}(0)+\kappa\mathcal{Q}(0)$
$\displaystyle\qquad=\mathcal{V}(0)+\kappa\Big{(}C^{\ast}\mathcal{V}^{2}(0)+\mathcal{V}(0)+\widetilde{K_{b0}}\sum_{l=0}^{\infty}\theta_{l}\Big{)}$
$\displaystyle\qquad\leq
C\Big{(}\mathcal{V}(0)+\sum_{l=0}^{\infty}\theta_{l}\Big{)}\ll 1.$

This implies that $\mathcal{V}(x)\ll 1$ for all $x>0$, since $C$ is
independent of $x$.

Furthermore, the total variation of $U^{\vartheta}(x,\cdot)$ is uniformly
bounded. More precisely, we conclude that

${\rm TV}\\{U^{\vartheta}(x,\cdot)\\}\approx
V(x)+|U^{\diamond}(x)-U_{0}^{+}|+|U_{\diamond}(x)-U_{0}^{-}|+|\sigma_{20}|+|\sigma_{30}|=\mathcal{O}(1).$
(3.4)

This completes the proof.

In order to have a front tracking approximate solution
$U^{\vartheta}(x,\cdot)$ defined for any time $x>0$, along with a uniform
bound on the total variation, we also need to have that the number of wave-
fronts in $U^{\vartheta}(x,\cdot)$ is finite. This is given by the subsequent
lemma.

Lemma 3.2. _For any fixed $\vartheta>0$ small enough, the number of wave-
fronts in $U^{\vartheta}\left(x,y\right)$ is finite and the approximate
solutions $U^{\vartheta}\left(x,y\right)$ are defined for all $x>0$. Moreover,
for any $x>0$, the total strength of the all non-physical waves is of order
$\mathcal{O}(1)\left(\delta_{\vartheta}+\omega_{\vartheta}\right)$_.

Proof. We first note the total interaction potential $\mathcal{Q}(x)$ remains
unchanged when there is no interaction and decreases across an interaction
“time” $x>0$ as discussed in Cases 1–5 in Proposition 3.1. Furthermore, from
Cases 1–5 and the subsequent analysis above, we have concluded that
$\mathcal{V}(x)\ll 1$. Hence, one can fix some number $\nu\in(0,1)$ such that

$\displaystyle\Delta\mathcal{Q}(x)$ $\displaystyle=$
$\displaystyle\mathcal{Q}(x^{+})-\mathcal{Q}(x^{-})$ (3.8) $\displaystyle\leq$
$\displaystyle\left\\{\begin{array}[]{ll}-\nu|b_{\alpha}b_{\beta}|&\mbox{ if
both waves $\alpha$ and $\beta$ are weak,}\\\ -\nu|b_{\alpha}|&\mbox{ if the
weak wave $\alpha$ hits the strong vortex sheet/entropy wave,}\\\
-\nu|\theta_{l}|&\mbox{ if the angle of the boundary
changes.}\end{array}\right.$

Now, following an argument similar to the one given in [2], we reach the
following conclusions. Note that initially $\mathcal{Q}(0)$ is bounded and $Q$
decreases thereafter for each case. Moreover, in the case where the
interaction potential between the incoming waves or the change of the angle of
the boundary is larger than $\omega_{\vartheta}$, $Q$ decreases by at least
$\nu\omega_{\vartheta}$ in these interactions, as implied by the bounds given
in (3.8). Following the wave-front tracking method in our problem, new
physical waves can be only produced by such interactions. Furthermore, when
the weak wave $\alpha$ of 1-family collides with the wall boundary, we solve
the lateral Riemann problem and have shown earlier that, after this
interaction, there is only a reflected wave of 4-family with the reflection
coefficient $1$. Hence, before and after this interaction, the number of the
waves keeps the same, and this implies that the number of the waves is finite.
Finally, because non-physical waves are generated only when physical waves
collide, we can also conclude that the number of non-physical wave fronts are
finite; and, provided that two waves can only collide once, the number of
interactions is also finite. Consequently, it follows that the approximate
solutions $U^{\vartheta}(x,\cdot)$ are defined for all times $x>0$. The
similar argument allows us to conclude that the total strength of all non-
physical wave fronts at any $x$ is of order
$\mathcal{O}(1)(\delta_{\vartheta}+\omega_{\vartheta})$. This completes the
proof Lemma 3.2.

Following the line of arguments given in [2, 4] for the wave-front tracking
algorithm and Lemma 3.1 above, we finish this section with the following
theorem for the global existence of entropy solutions to the initial-boundary
value problem (1.1) and (1.7)–(1.8).

Theorem 3.1. _Suppose that ${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm
TV}(g^{\prime}(\cdot))$ is small enough. Then, for the initial-boundary value
problem (1.1) and (1.7)–(1.8), there exists a global weak solution in BV
satisfying the steady Clausius entropy inequality (1.9)._

##  4\. The Lyapunov Functional for the ${\bf L}^{1}$–Distance between Two
Solutions

To show that the front tracking approximations, constructed for the existence
analysis in Section 3, converge to a unique limit, we estimate the distance
between any two $\vartheta$-approximate $U$ and $V$ of problem (1.1) and
(1.7)–(1.8) of initial-boundary value type. To this end, we develop the
Lyapunov functional $\Phi(U,V)$, equivalent to the $L^{1}$–distance:

${C}^{-1}\,\lVert
U(x,\cdot)-V(x,\cdot)\rVert_{L^{1}}\leq\mathit{\Phi}(U,V)\leq{C}\,\|U(x,\cdot)-V(x,\cdot)\|_{L^{1}},$

and prove that the functional $\Phi(U,V)$ is almost decreasing along pairs of
solutions,

$\mathit{\Phi}\left(U(x_{2},\cdot),V(x_{2},\cdot)\right)-\mathit{\Phi}\left(U(x_{1},\cdot),V(x_{1},\cdot)\right)\leq
C\vartheta(x_{2}-x_{1}),\hskip 14.22636pt\text{for all }x_{2}>x_{1}>0,$

for some constant ${C}>0$. Here $U$ and $V$ are two approximate solutions
constructed via the wave-front tracking method, and the small approximation
parameter $\vartheta$ is responsible for controlling the subsequent errors:

  * •

Errors in the approximation of the initial data and the boundary.

  * •

Errors in the speeds of shock, vortex sheet, entropy wave, and rarefaction
fronts.

  * •

The total strength of all non-physical fronts.

  * •

The maximum strength of rarefaction fronts.

Along the line of arguments presented in [9, 22, 24], with “time” $x$ fixed,
at each $y$, one connects the state $U(y)$ with $V(y)$ in the state space by
going along the Hugoniot curves $S_{1},C_{2},C_{3}$, and $S_{4}$. Depending on
the location of the strong vortex sheet/entropy wave in $U(y)$ and $V(y)$, the
distance between $U(y)$ and $V(y)$ is estimated along discontinuity waves in
possibly different “directions”, determining the strength of the $j$-Hugoniot
wave $h_{j}(y)$ in the following way:

  * •

Suppose that $U(y)$ and $V(y)$ are both in $\Omega_{-}$ and $\Omega_{+}$. Then
one begins at the state $U(y)$ and moves along the Hugoniot curves to reach
the state $V(y)$.

  * •

Suppose that $U(y)$ is in $\Omega_{-}$ and $V(y)$ is in $\Omega_{+}$. Then one
begins at the state $U(y)$ and moves along the Hugoniot curves to reach the
state $V(y)$.

  * •

Suppose that $V(y)$ is in $\Omega_{-}$ and $U(y)$ is in $\Omega_{+}$. Then one
begins at the state $V(y)$ and moves along the Hugoniot curves to reach the
state $U(y)$.

Define the $L^{1}$–weighted strengths of the waves in the solution of the
Riemann problem $\left(U(y),V(y)\right)$ or $\left(V(y),U(y)\right)$ as
follows:

$\displaystyle q_{j}(y)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}w^{b}_{j}\cdot h_{j}(y)&\mbox{
whenever $U(y)$ and $V(y)$ are both in $\Omega_{-}$,}\\\ w^{m}_{j}\cdot
h_{j}(y)&\mbox{ whenever $U(y)$ and $V(y)$ are both in different domains,}\\\
w^{a}_{j}\cdot h_{j}(y)&\mbox{ whenever $U(y)$ and $V(y)$ are both in
$\Omega_{+}$,}\end{array}\right.$ (4.4)

with the constants $w^{b}_{j}$, $w^{m}_{j}$, and $w^{a}_{j}$ above to be
specified later on, based on the estimates of wave interactions and
reflections in Lemmas 2.2–2.7.

We define the following Lyapunov functional,

$\mathit{\Phi}(U,V)=\sum_{j=1}^{4}\int\limits_{g(x)}^{\infty}\
|q_{j}(y)|W_{j}(y)\,dy,$ (4.5)

where the weights are given by

$W_{j}(y)=1+\kappa_{1}A_{j}(y)+\kappa_{2}\left(\mathcal{Q}(U)+\mathcal{Q}(V)\right).$
(4.6)

The constants $\kappa_{1}$ and $\kappa_{2}$ are to be determined later. Here
$\mathcal{Q}$ denotes the total wave interaction potential incorporating the
boundary effect as defined in (3.2), and $A_{j}(y)$ denotes the total strength
of waves in $U$ and $V$, which approach the $j$-wave $q_{j}(y)$, defined in
the following manner (for $y$ where there is no jump in $U$ or $V$):

$A_{j}(y)=F_{j}(y)+G_{j}(y)+\left\\{\begin{array}[]{ll}H_{j}(y)&\mbox{ if
$j$-wave $q_{j}(y)$ is small and the $j$-field is genuinely nonlinear,}\\\
0&\mbox{ if $j$ = 2, 3 and $q_{j}(y)$ = $B$ is large.}\end{array}\right.$
(4.7)

Next, we define the following global weights $G_{j}$:

$G_{j}(y)=$ |  $U,V$ are both in $\Omega_{-}$ |  $U,V$ are in distinct regions |  $U,V$ are both in $\Omega_{+}$
---|---|---|---
$G_{1}(y)$ | 4B | 2B | 4B
$G_{2,3}(y)$ | 0 | 0 | 0
$G_{4}(y)$ | 4B | 2B | 2B

Under the assumption that ${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm
TV}(\widetilde{V_{0}}(\cdot))+{\rm TV}(g^{\prime}(\cdot))$ is small enough
with $U(x,\cdot)$, $V(x,\cdot)$ $\in{\rm BV}\cap L^{1}$, one concludes

$\displaystyle\mathcal{M}^{-1}\lVert
U(x,\cdot)-V(x,\cdot)\rVert_{L^{1}}\leq\sum_{j=1}^{4}\int\limits^{\infty}\limits_{g(x)}|q_{j}(y)|\,dy\leq\mathcal{M}\|U(x,\cdot)-V(x,\cdot)\|_{L^{1}},$
$\displaystyle 1\leq W_{j}(y)\leq\mathcal{M},\hskip 5.69054ptj=1,\ldots,4,$

where the constant $\mathcal{M}$ is independent of $\vartheta$ and “time” $x$.
Here _we define the strength of any large wave of the $2$\- or
$3$-characteristic family to equal to some fixed number B (bigger than all
strengths of small waves),_ and the concepts “small” and “large” mean the
waves that connect the states in the same or in the distinct domains
$\Omega^{-}$ and $\Omega^{+}$, respectively.

The summands in (4.7) are defined as follows,

$\displaystyle F_{j}(y)$ $\displaystyle=$
$\displaystyle\Biggl{(}\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}\setminus\mathcal{SC}\\\
y_{\alpha}<y,j<k_{\alpha}\leq
4\end{subarray}}+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}\setminus\mathcal{SC}\\\
y_{\alpha}>y,1\leq k_{\alpha}<j\end{subarray}}\Biggr{)}|\alpha|,$
$\displaystyle H_{j}(y)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}(\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(U)\setminus\mathcal{SC},y_{\alpha}<y,k_{\alpha}=j\end{subarray}}+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(V)\setminus\mathcal{SC},y_{\alpha}>y,k_{\alpha}=j\end{subarray}})|\alpha|&\quad\mbox{
if $q_{j}(y)<0$,}\\\
(\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(V)\setminus\mathcal{SC},y_{\alpha}<y,k_{\alpha}=j\end{subarray}}+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(U)\setminus\mathcal{SC},y_{\alpha}>y,k_{\alpha}=j\end{subarray}})|\alpha|&\quad\mbox{
if $q_{j}(y)>0$,}\end{array}\right.$ (4.10)

where, at each $x$, $\alpha$ stands for the (non-weighted) strength of the
wave $\alpha\in\mathcal{J}$, located at the point $y_{\alpha}$ and belonging
to the characteristic family $k_{\alpha}$;
$\mathcal{J}=\mathcal{J}(U)\cup\mathcal{J}(V)$,
$\mathcal{SC}=\mathcal{SC}(U)\cup\mathcal{SC}(V)$ is the set of all waves (in
$U$ and $V$) and the set of all large (strong) characteristic discontinuities
(in $U$ and $V$) respectively.

Consequently, there holds

${C}^{-1}\lVert
U(x,\cdot)-V(x,\cdot)\rVert_{L^{1}}\leq\mathit{\Phi}(U,V)\leq{C}\|U(x,\cdot)-V(x,\cdot)\|_{L^{1}},$
(4.11)

for any $x\geq 0$ with the constant ${C}>0$ depending only on the quantities
independent of $x$: the strength of the strong vortex sheet/entropy wave and
${\rm TV}(\widetilde{U_{0}}(\cdot))+{\rm TV}(\widetilde{V_{0}}(\cdot))+{\rm
TV}(g^{\prime}(\cdot))$.

We now analyze the evolution of the Lyapunov functional $\mathit{\Phi}$ in the
flow direction $x>0$. For $j=1,\ldots,4$, we call $\lambda_{j}(y)$ the speed
of the $j$-wave $q_{j}(y)$ (along the Hugoniot curve in the phase space).
Then, at a “time” $x>0$ which is not the interaction time of the waves either
in $U(x)=U(x,\cdot)$ or $V(x)=V(x,\cdot)$, an explicit computation gives

$\displaystyle{d\over dx}\mathit{\Phi}\left(U(x),V(x)\right)$
$\displaystyle=\sum_{\alpha\in\mathcal{J}}\sum_{j=1}^{4}\left(\lvert
q_{j}(y_{\alpha}^{-})\rvert W_{j}(y_{\alpha}^{-})-\lvert
q_{j}(y_{\alpha}^{+})\rvert
W_{j}(y_{\alpha}^{+})\right)\dot{y}_{\alpha}+\sum_{j=1}^{4}\lvert
q_{\j}(b)\rvert W_{j}(b)\dot{y}_{b}$
$\displaystyle=\sum_{\alpha\in\mathcal{J}}\sum_{j=1}^{4}\left(\lvert
q_{j}(y_{\alpha}^{-})\rvert
W_{j}(y_{\alpha}^{-})\left(\dot{y}_{\alpha}-\lambda_{j}(y_{\alpha}^{-})\right)-\lvert
q_{j}(y_{\alpha}^{+})\rvert
W_{j}(y_{\alpha}^{+})\left(\dot{y}_{\alpha}-\lambda_{j}(y_{\alpha}^{+})\right)\right)$
$\displaystyle\quad+\sum_{j=1}^{4}\lvert q_{j}(b)\rvert
W_{j}(b)\left(\dot{y}_{b}+\lambda_{j}(b)\right),$ (4.12)

where $\dot{y}_{\alpha}$ denotes the speed of the Hugoniot wave
$\alpha\in\mathcal{J}$, $b=g(x)^{+}$ stands for the points close to the
boundary, and $\dot{y}_{b}$ is the slope of the boundary.

We present the notation

$\displaystyle E_{\alpha,j}$ $\displaystyle=\lvert q_{j}^{+}\rvert
W_{j}^{+}\left(\lambda_{j}^{+}-\dot{y}_{\alpha}\right)-\lvert q_{j}^{-}\rvert
W_{j}^{-}\left(\lambda_{j}^{-}-\dot{y}_{\alpha}\right),$ (4.13) $\displaystyle
E_{b,j}$ $\displaystyle=\lvert q_{j}(b)\rvert
W_{j}(b)\left(\dot{y}_{b}+\lambda_{j}(b)\right),$ (4.14)

where $q_{j}^{\pm}$ = $q_{j}(y^{\pm}_{\alpha})$,
$W^{\pm}_{j}=W_{j}(y^{\pm}_{\alpha})$, and $\lambda^{\pm}_{j}$ =
$\lambda_{j}(y^{\pm}_{\alpha})$.

Then (4.12) can be written as

${d\over
dx}\mathit{\Phi}\left(U(x),V(x)\right)=\sum_{\alpha\in\mathcal{J}}\sum_{j=1}^{4}E_{\alpha,j}+\sum_{j=1}^{4}E_{b,j}.$
(4.15)

Our central aim is to prove the bounds:

$\displaystyle\sum_{j=1}^{4}E_{\alpha,j}\leq\mathcal{O}(1)\vartheta\lvert\alpha\rvert\quad\text{
when $\alpha$ is a weak wave in $\mathcal{J}$},$ (4.16)
$\displaystyle\sum_{j=1}^{4}E_{\alpha,j}\leq\mathcal{O}(1)\lvert\alpha\rvert\quad\text{
when $\alpha$ is a non-physical wave in $\mathcal{J}$,}$ (4.17)
$\displaystyle\sum_{j=1}^{4}E_{\alpha,j}\leq 0\quad\text{ when $\alpha$ is a
strong vortex sheet/entropy wave in $\mathcal{J}$,}$ (4.18)
$\displaystyle\sum_{j=1}^{4}E_{b,j}\leq 0\quad\text{ near the boundary},$
(4.19)

where the quantities denoted by the Landau symbol $\mathcal{O}$(1) are
independent of the constants $\kappa_{1}$ and $\kappa_{2}$.

From (4.16)–(4.19) together with the uniform bound on the total strengths of
waves (3.4), we obtain

${d\over dx}\mathit{\Phi}\left(U(x),V(x)\right)\leq\mathcal{O}(1)\vartheta.$
(4.20)

Integration of (4.20) over the interval $\left[0,x\right]$ yields

$\mathit{\Phi}\left(U(x),V(x)\right)\leq\mathit{\Phi}\left(U(0),V(0)\right)+\mathcal{O}(1)\vartheta
x.$ (4.21)

We remark that, at each interaction “time” $x$ when two fronts of $U$ or two
fronts of $V$ interact, by the Glimm interaction estimates, all the weight
functions $W_{j}(y)$ decrease, if the constant $\kappa_{2}$ in the Lyapunov
functional is taken to be sufficiently large. Furthermore, due to the self-
similar property of the Riemann solutions, $\mathit{\Phi}$ decreases at this
“time”.

In the next section, we establish the bounds (4.16)–(4.19), particularly
(4.18) and (4.19), when $\alpha$ is a strong vortex sheet/entropy wave in
$\mathcal{J}$ and near the Lipschitz wall boundary, respectively.

##  5\. The $L^{1}$–Stability Estimates

For the case of the non-physical waves in $\mathcal{J}$, as well as the case
that the weak wave
$\alpha\in\mathcal{J}\mathrel{\mathop{:}}=\mathcal{J}(U)\cup\mathcal{J}(V)$,
which appears when $U$ and $V$ are both in $\Omega_{-}$ or $\Omega_{+}$,
estimates (4.16) and (4.17) are shown similarly based on the arguments in
Bressan-Liu-Yang [9], provided that $\frac{2|B|}{|\sigma_{20}|+|\sigma_{30}|}$
is sufficiently small and $\kappa_{1}$ is sufficiently large. In what follows,
we focus only on the other two cases, namely (4.18) and (4.19).

Case 1: The first strong vortex sheet/entropy wave $\alpha$ in $U$ or $V$ is
crossed. Then, by Lemma 2.4, we have the estimates:

$\displaystyle h^{+}_{1}$ $\displaystyle=$ $\displaystyle
h^{-}_{1}+K_{11}h^{-}_{4},$ (5.1) $\displaystyle h^{+}_{4}$ $\displaystyle=$
$\displaystyle K_{14}h^{-}_{4}.$ (5.2)

Moreover, the essential estimate $|K_{11}|<1$ given in Lemma 2.4 ensures the
existence of desired weights $w^{b}_{1}$ and $w^{b}_{4}$ in the following way.

Lemma 5.1. _There exist $w^{b}_{1}$, $w^{b}_{4}$, and $\gamma_{b}$ satisfying_

$\displaystyle\frac{w^{b}_{4}}{w^{b}_{1}}<1,$ (5.3)
$\displaystyle\frac{w^{b}_{1}}{w^{b}_{4}}K_{11}\left|\frac{\lambda_{1}^{-}-\lambda_{2,3}}{\lambda_{4}^{-}-\lambda_{2,3}}\right|<\gamma_{b}<1.$
(5.4)

With Lemma 5.1, we estimate $E_{j}$ for $j=1,\ldots,4$, starting with $E_{1}$:
By (5.1) and (5.4),

$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle|q_{1}^{-}|(\lambda_{1}^{-}-\dot{y}_{\alpha})(W_{1}^{+}-W_{1}^{-})+W^{+}_{1}\left(|q^{+}_{1}|(\lambda_{1}^{+}-\dot{y}_{\alpha})-|q^{-}_{1}|(\lambda^{-}_{1}-\dot{y}_{\alpha})\right)$
$\displaystyle=$ $\displaystyle
2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+W^{+}_{1}\left(|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle\leq$ $\displaystyle
2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+W^{+}_{1}\left(w^{b}_{1}|h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+w^{b}_{1}K_{11}|h^{-}_{4}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle\leq$ $\displaystyle
2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+2B\kappa_{1}\left(w^{b}_{1}|h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+\gamma_{b}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})-w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle+(\kappa_{1}A_{W^{+}_{1}}+M)|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-(\kappa_{1}A_{W^{+}_{1}}+M)|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|,$

where
$W^{+}_{1}=W_{1}(y_{\alpha}^{+})=2B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M$, and
$M=1+\kappa_{2}(\mathcal{Q}(U)+\mathcal{Q}(V))$ is a positive constant. The
term $A_{W^{+}_{1}}=F_{1}(y_{\alpha}^{+})+H_{1}(y_{\alpha}^{+})$ here is the
total strength of all the weak waves in $U$ and $V$ which approach the
$1$-wave $q_{1}^{+}=q_{1}(y_{\alpha}^{+})$, and the term $2B\kappa_{1}$ is
from the weight $G_{1}(y_{\alpha}^{+})$.

For $j=2,3$, since $W^{+}_{j}=W^{-}_{j}$, (4.12) reduces to

$\displaystyle
E_{j}=W^{-}_{j}\big{(}|q_{j}^{+}|(\lambda_{j}^{+}-\dot{y}_{\alpha})-|q_{j}^{-}|(\lambda_{j}^{-}-\dot{y}_{\alpha})\big{)}\leq\mathcal{O}(1)B\Big{(}\vartheta+\sum_{i\neq\\{{2,3\\}}}|q_{i}^{-}|+|q_{k}^{-}|\Big{)},$

where $k\neq\\{j,1,4\\}$.

For $j$ = 4,

$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle|q_{4}^{-}|(\lambda_{4}^{-}-\dot{y}_{\alpha})(W_{4}^{+}-W_{4}^{-})+W^{+}_{4}\left(|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle=$
$\displaystyle-2B\kappa_{1}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})+\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})$
$\displaystyle-{}\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha}),$

where
$W^{+}_{4}=W_{4}(y_{\alpha}^{+})=2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M$, and
$M=1+\kappa_{2}(\mathcal{Q}(U)+\mathcal{Q}(V))$ is a positive constant. The
term $A_{W^{+}_{4}}=F_{4}(y_{\alpha}^{+})+H_{4}(y_{\alpha}^{+})$ is the total
strength of all the weak waves in $U$ and $V$ which approach the $4$-wave
$q_{4}^{+}=q_{4}(y_{\alpha}^{+})$, and the term $2B\kappa_{1}$ is from the
weight $G_{4}(y_{\alpha}^{+})$.

For the weighted $L^{1}$–strength $q_{j}(y)$ in (4.4), we choose $w^{b}_{1}$
small enough relatively to $w^{m}_{1}$ and $w^{b}_{4}$ large enough relatively
to $w^{m}_{4}$, choose $\kappa_{1}$ large enough and the total variation of
$U$ and $V$ so small, and use (5.1)–(5.2) to obtain

$\displaystyle\sum^{4}_{j=1}E_{j}$ $\displaystyle\leq$ $\displaystyle
2B\kappa_{1}\left(w^{b}_{1}|h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+\gamma_{b}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})-w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\right)$
$\displaystyle+{}(\kappa_{1}A_{W^{+}_{1}}+M)|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-(\kappa_{1}A_{W^{+}_{1}}+M)|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+2B\kappa_{1}w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}w^{m}_{4}|K_{14}p^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})$
$\displaystyle-{}\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{4}}+M\big{)}|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{-}|+\left(|q_{2}^{-}|+|q_{3}^{-}|\right)\Big{)}$
$\displaystyle=$
$\displaystyle-(1-\gamma_{b})2B\kappa_{1}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+(\kappa_{1}A_{W^{+}_{4}}+M)\big{(}w^{m}_{4}|K_{14}h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\big{)}$
$\displaystyle+2B\kappa_{1}w^{b}_{1}(|h^{+}_{1}|+|h^{-}_{1}|)|\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+2B\kappa_{1}w^{m}_{4}|K_{14}h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-2B\kappa_{1}w^{b}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+(\kappa_{1}A_{W^{+}_{1}}+M)w^{b}_{1}|h^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-(\kappa_{1}A_{W^{+}_{1}}+M)w^{m}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+{}\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{-}|+\left(|q_{2}^{-}|+|q_{3}^{-}|\right)\Big{)}$
$\displaystyle\leq$ $\displaystyle 0.$

Case 2: The weak wave $\alpha$ between the two strong vortex sheets/entropy
waves in $U$ and $V$ is crossed. For $j=1$, we have

$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle|q^{\pm}_{1}|(W^{+}_{1}-W^{-}_{1})(\lambda^{\pm}_{1}-\dot{y}_{\alpha})+W^{\mp}_{1}\left(|q^{+}_{1}|(\lambda^{+}_{1}-\dot{y}_{\alpha})-|q^{-}_{1}|(\lambda^{-}_{1}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{1}||\alpha||\lambda^{\pm}_{1}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{1}|-|q^{-}_{1}|)(\lambda^{+}_{1}-\dot{y}_{\alpha})+|q^{-}_{1}|(\lambda^{+}_{1}-\lambda^{-}_{1})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{1}||\alpha||\lambda^{\pm}_{1}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{1}|-|q^{-}_{1}|)(\lambda^{+}_{1}-\dot{y}_{\alpha})+\mathcal{O}(1)|q^{-}_{1}||\alpha|\right).$

For the cases when $j=2,3$, we have

$\displaystyle E_{j}$ $\displaystyle=$ $\displaystyle
B\left(\big{(}W^{+}_{j}-W^{-}_{j}\big{)}\big{(}\lambda^{\pm}_{j}-\dot{y}_{\alpha}\big{)}+W^{\mp}_{j}\big{(}\lambda^{\pm}_{j}-\lambda^{\mp}_{j}\big{)}\right)$
$\displaystyle\leq$ $\displaystyle
B\left(-\kappa_{1}|\alpha||\lambda^{+}_{j}-\dot{y}_{\alpha}|+\mathcal{O}(1)|\alpha|\right).$

For $j$ = 4, we have

$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle|q^{\pm}_{4}|(W^{+}_{4}-W^{-}_{4})(\lambda^{\pm}_{4}-\dot{y}_{\alpha})+W^{\mp}_{4}\left(|q^{+}_{4}|(\lambda^{+}_{4}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{4}||\alpha||\lambda^{\pm}_{4}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{4}|-|q^{-}_{4}|)(\lambda^{+}_{4}-\dot{y}_{\alpha})+|q^{-}_{4}|(\lambda^{+}_{4}-\lambda^{-}_{4})\right)$
$\displaystyle\leq$
$\displaystyle\kappa_{1}|q^{\pm}_{4}||\alpha||\lambda^{\pm}_{4}-\dot{y}_{\alpha}|+2B\kappa_{1}\left((|q^{+}_{4}|-|q^{-}_{4}|)(\lambda^{+}_{4}-\dot{y}_{\alpha})+\mathcal{O}(1)|q^{-}_{4}||\alpha|\right).$

Then we have

$\sum_{j=1}^{4}E_{j}\leq\kappa_{1}\mathcal{O}(1)\Big{(}-|\alpha|+|\alpha|\sum_{k\neq\\{2,3\\}}|q_{k}^{+}|+|q_{k}^{-}|+\sum_{k\neq\\{2,3\\}}|q_{k}^{+}|-|q_{k}^{-}|\Big{)}+\mathcal{O}(1)|\alpha|.$

Since
$\big{|}|q_{k}^{+}|-|q_{k}^{-}|\big{|}\leq|q_{k}^{+}-q_{k}^{-}|\leq\mathcal{O}(1)|\alpha|$
when $k\neq\\{2,3\\}$, we obtain

$\sum_{j=1}^{4}E_{j}\leq 0$

if all the weights $w^{m}_{j}$ are small enough and $\kappa_{1}$ is
sufficiently large.

Notice that the choice of the upper or lower superscripts depends on the
family number $k_{\alpha}$.

Case 3: The second strong vortex sheet/entropy wave $\alpha$ in $U$ or $V$ is
crossed. For this case, by Lemma 2.6, we have

$\displaystyle h^{-}_{1}$ $\displaystyle=$ $\displaystyle K_{21}h^{+}_{1},$
(5.5) $\displaystyle h^{-}_{4}$ $\displaystyle=$ $\displaystyle
h^{+}_{4}+K_{24}h^{+}_{1}.$ (5.6)

Moreover, the essential estimate $|K_{24}|<1$ in Lemma 2.6 ensures the
existence of desired weights $w^{a}_{1}$ and $w^{a}_{4}$ in the following
manner.

Lemma 5.2. _There exist $w^{a}_{1}$, $w^{a}_{4}$, and $\gamma_{a}$ satisfying_

$\frac{w^{a}_{4}}{w^{a}_{1}}\left|\frac{\lambda_{4}^{+}-\lambda_{2,3}}{\lambda_{1}^{+}-\lambda_{2,3}}\right|K_{24}<\gamma_{a}<1.$
(5.7)

With Lemma 5.2, we estimate $E_{j}$ for $j=1,\ldots,4$ as follows: By (5.5),

$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle|q_{1}^{-}|(\lambda_{1}^{-}-\dot{y}_{\alpha})(W_{1}^{+}-W_{1}^{-})+W^{+}_{1}\left(|q^{+}_{1}|(\lambda_{1}^{+}-\dot{y}_{\alpha})-|q^{-}_{1}|(\lambda^{-}_{1}-\dot{y}_{\alpha})\right)$
$\displaystyle=$
$\displaystyle-2B\kappa_{1}|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+(4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M)|q^{+}_{1}|(\lambda_{1}^{+}-\dot{y}_{\alpha})$
$\displaystyle{}+w^{m}_{1}|K_{21}h^{+}_{1}|(4B\kappa_{1}+A_{W^{+}_{1}}+M)|\lambda_{1}^{-}-\dot{y}_{\alpha}|,$
$\displaystyle=$
$\displaystyle-2B\kappa_{1}|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|+w^{m}_{1}|K_{21}h^{+}_{1}|(4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M)|\lambda_{1}^{-}-\dot{y}_{\alpha}|$
$\displaystyle-
w_{1}^{a}|h^{+}_{1}|(2B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M)|\lambda_{1}^{+}-\dot{y}_{\alpha}|-2B\kappa_{1}w_{1}^{a}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|,$

where
$W^{+}_{1}=W_{1}(y_{\alpha}^{+})=4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M$, and
$M=1+\kappa_{2}\big{(}\mathcal{Q}(U)+\mathcal{Q}(V)\big{)}$ is a positive
constant. The term $A_{W^{+}_{1}}=F_{1}(y_{\alpha}^{+})+H_{1}(y_{\alpha}^{+})$
here is the total strength of all the weak waves in $U$ and $V$ which approach
the $1$-wave $q_{1}^{+}=q_{1}(y_{\alpha}^{+})$, and the term $4B\kappa_{1}$ is
from the weight $G_{1}(y_{\alpha}^{+})$.

For $j=2,3$, since $W^{+}_{j}=W^{-}_{j}$, (4.12) reduces to

$\displaystyle E_{j}$ $\displaystyle=$ $\displaystyle
W^{+}_{j}\big{(}|q_{j}^{+}|(\lambda_{j}^{+}-\dot{y}_{\alpha})-|q_{j}^{-}|(\lambda_{j}^{-}-\dot{y}_{\alpha})\big{)}$
$\displaystyle\leq$
$\displaystyle\mathcal{O}(1)B\Big{(}\vartheta+\sum_{i\neq\\{{2,3\\}}}|q_{i}^{+}|+|q_{k}^{+}|\Big{)},$

where $k\neq\\{j,1,4\\}$.

By (5.6) and (5.7),

$\displaystyle E_{4}$ $\displaystyle=$
$\displaystyle|q_{4}^{-}|(\lambda_{4}^{-}-\dot{y}_{\alpha})(W_{4}^{+}-W_{4}^{-})+W^{+}_{4}\left(|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle=$ $\displaystyle
W^{+}_{4}\left(|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-|q^{-}_{4}|(\lambda^{-}_{4}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$ $\displaystyle
W^{+}_{4}\left(w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})+w^{a}_{4}K_{24}|h^{+}_{1}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\right)$
$\displaystyle\leq$ $\displaystyle
2B\kappa_{1}\left(w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})+\gamma_{a}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|-w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\right)$
$\displaystyle+(\kappa_{1}A_{W^{+}_{4}}+M)|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-(\kappa_{1}A_{W^{+}_{4}}+M)|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha}),$

where
$W^{+}_{4}=W_{4}(y_{\alpha}^{+})=2\kappa_{1}B+\kappa_{1}A_{W^{+}_{4}}+M$, and
$M=1+\kappa_{2}(\mathcal{Q}(U)+\mathcal{Q}(V))$ is a positive constant. The
term $A_{W^{+}_{4}}=F_{4}(y_{\alpha}^{+})+H_{4}(y_{\alpha}^{+})$ here is the
total strength of all the weak waves in $U$ and $V$ which approach the
$4$-wave $q_{4}^{+}=q_{4}(y_{\alpha}^{+})$, and the term $2B\kappa_{1}$ is
from the weight $G_{4}(y_{\alpha}^{+})$.

For the weighted $L^{1}$–strength $q_{j}(y)$ in (4.1), when $w^{a}_{4}$ is
small enough relatively to $w^{m}_{4}$ and $w^{a}_{1}$ is large enough
relatively to $w^{m}_{1}$, $\kappa_{1}$ is large enough, applying (5.5) and
(5.6), the total variation of $u$ and $v$ is so small that

$\displaystyle\sum^{4}_{j=1}E_{j}$ $\displaystyle\leq$ $\displaystyle
2B\kappa_{1}\left(w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})+\gamma_{a}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|-w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})\right)$
$\displaystyle+{}(\kappa_{1}A_{W^{+}_{4}}+M)|q^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-(\kappa_{1}A_{W^{+}_{4}}+M)|q^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle-2B\kappa_{1}|q^{-}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}|q^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+\big{(}4B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M\big{)}w^{m}_{1}|K_{21}h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|$
$\displaystyle-{}\big{(}2B\kappa_{1}+\kappa_{1}A_{W^{+}_{1}}+M\big{)}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{+}|+\left(|q_{2}^{+}|+|q_{3}^{+}|\right)\Big{)}$
$\displaystyle=$
$\displaystyle-(1-\gamma_{a})2B\kappa_{1}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+(\kappa_{1}A_{W^{+}_{1}}+M)\big{(}w^{m}_{1}|K_{21}h^{+}_{1}||\lambda_{1}^{-}-\dot{y}_{\alpha}|-w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|\big{)}$
$\displaystyle+2B\kappa_{1}w^{a}_{4}|h^{-}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-2B\kappa_{1}w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+2B\kappa_{1}(-w^{m}_{1}|K_{21}h^{+}_{1}|+2w^{m}_{1}|K_{21}h^{+}_{1}|)|\lambda_{1}^{-}-\dot{y}_{\alpha}|-2B\kappa_{1}w^{a}_{1}|h^{+}_{1}||\lambda_{1}^{+}-\dot{y}_{\alpha}|$
$\displaystyle+(\kappa_{1}A_{W^{+}_{4}}+M)w^{a}_{4}|h^{+}_{4}|(\lambda_{4}^{+}-\dot{y}_{\alpha})-(\kappa_{1}A_{W^{+}_{4}}+M)w^{m}_{4}|h^{-}_{4}|(\lambda_{4}^{-}-\dot{y}_{\alpha})$
$\displaystyle+{}\mathcal{O}(1)B\Big{(}2\vartheta+2\sum_{i\neq\\{2,3\\}}|q_{i}^{+}|+\left(|q_{2}^{+}|+|q_{3}^{+}|\right)\Big{)}$
$\displaystyle\leq$ $\displaystyle 0,$

which yields (4.16).

Case 4: Close to the Lipschitz wall boundary. This case differs from the
Cauchy problem. Here we will use the particular property of the boundary
condition (1.8): The flows of $U$ and $V$ are tangent to the Lipschitz wall,
implying that they must be parallel with each other along the boundary. Then a
piecewise constant weak solution is constructed only along the Hugoniot curves
determined by the Riemann data $U(b)$ and $V(b)$, the states of solutions $U$
and $V$, respectively, close to the boundary.

Proposition 5.2. _Suppose that U $\left(b\right)$ =
$(\breve{u},\breve{v},\breve{p},\breve{\rho})$ and V$\left(b\right)$ =
$(\tilde{u},\tilde{v},\tilde{p},\tilde{\rho})$ are states in a small
neighborhood $O_{\varepsilon}(U_{-})$ of $U_{-}$ satisfying
$\frac{\breve{v}}{\breve{u}}=\frac{\tilde{v}}{\tilde{u}}=\dot{z}_{b}$, and
$\breve{v},\tilde{v}\approx 0$. Denote by $h_{j}(b)$ the strength of the
$j^{th}$ shock in the Riemann problem determined by U$\left(b\right)$ and
V$\left(b\right)$, and by $\lambda_{j}$ the corresponding
$j^{th}$-characteristic speed. Then_

$\displaystyle|\lambda_{j}-\dot{z}_{b}|\sim|h_{1}(b)|,\hskip 5.69054ptj=2,3,$
(5.8)
$\displaystyle|h_{4}(b)|\leq|h_{1}(b)|+\mathcal{O}(1)|h_{2}(b)||\lambda_{2}-\dot{z}_{b}|+|h_{1}(b)|\mathcal{O}(1)|\dot{z}_{b}|,$
(5.9) $\displaystyle|h_{1}(b)|=\bar{\mathcal{O}}(1)|h_{4}(b)|,\hskip
14.22636pt\frac{1}{2}<\tilde{\mathcal{O}}(1)<\frac{3}{2},$ (5.10)

_where $\dot{z}_{b}$ is the slope of the Lipschitz wall._

Proof. We do the proof by analyzing the following two cases.

_Case 1: $h_{1}(b)=0$ and $h_{4}(b)=0$ that corresponds to the case
$\breve{p}=\tilde{p}$_. Starting at the state $U_{b}$, we move along the
Hugoniot curves of the second and third families to reach $V_{b}$. Note that
these two families are the contact Hugoniot curves, and so $\lambda_{2}$ and
$\lambda_{3}$ are constant along the Hugoniot curves. Given that
$\lambda_{2,3}=\frac{v}{u}$, $\textbf{r}_{2}=(1,\frac{v}{u},0,0)^{\top}$, and
$\textbf{r}_{3}=(0,0,0,\rho)^{\top}$, the quantity $\frac{v}{u}$ remains
unchanged as the initial value $\frac{v(U_{b})}{u(U_{b})}$, i.e.,
$\dot{z}_{b}$ in this process by the boundary condition (1.8). Therefore, we
conclude that $\lambda_{2,3}=\dot{z}_{b}$, equivalently,

$\dot{z}_{b}-\lambda_{2,3}=0.$

_Case 2: $h_{1}(b)\neq 0$ that corresponds to $\breve{p}\neq\tilde{p}$_.
Starting at the state $U(b)$, we move along the 1-Hugoniot curve to reach
$U_{1}$, then possibly move along the 2-contact Hugoniot curve to reach
$U_{2}$, the 3-Hugoniot curve to reach $U_{3}$, and the 4-Hugoniot curve to
reach $V(b)$.

To make clear some essential relations among the strengths
$h_{1}(b),h_{2}(b),h_{3}(b),\text{and }h_{4}(b)$, we project $(u,v,p,\rho)$
onto the $(u,v)$–plane. Let $\textbf{r}_{1}|_{u}$ be the projection of
$\textbf{r}_{1}$ onto the $u$-axis, $\textbf{r}_{2}|_{(u,v)}$ be the
projection of $\textbf{r}_{2}$ onto the $(u,v)$–plane; and so on. At the
background state $U_{-}$, there holds

$\textbf{r}_{1}|_{u}=-\textbf{r}_{4}|_{u},\hskip
5.69054pt\textbf{r}_{1}|_{v}=\textbf{r}_{4}|_{v},\hskip
5.69054pt\textbf{r}_{1}|_{(p,\rho)}=-\textbf{r}_{4}|_{(p,\rho)},\hskip
5.69054pt$ $\textbf{r}_{2}=\textbf{r}_{2}|_{(u,v)},\hskip
5.69054pt\textbf{r}_{3}|_{(u,v)}=0.$

We first note that $h_{4}(b)\neq 0$. Given that
$\textbf{r}_{1}|_{(u,v)}=k_{1}(-\lambda_{1},1)^{\top}$ along with finite
characteristic speeds $\lambda_{1}$ and $\dot{z}_{b}\approx 0$, there always
holds $\dot{z}_{b}<-\frac{1}{\lambda_{1}}$ near the state $U_{-}$. So we can
conclude that, in the $(u,v)$–plane, the derivative $\frac{dv}{du}$ along the
1-curve is always larger than $\dot{z}_{b}$. This implies that
$\frac{v(U_{1})}{u(U_{1})}\neq\frac{v(U_{b})}{u(U_{b})}$. Meanwhile, we have
$\frac{v(U_{1})}{u(U_{1})}=\frac{v(U_{2})}{u(U_{2})}=\frac{v(U_{3})}{u(U_{3})}$
and $\frac{v(V_{b})}{u(V_{b})}=\frac{v(U_{b})}{u(U_{b})}$. Hence,

$\frac{v(U_{1})}{u(U_{1})}=\frac{v(U_{2})}{u(U_{2})}=\frac{v(U_{3})}{u(U_{3})}\neq\frac{v(V_{b})}{u(V_{b})}.$

Thus, we conclude that there is some distance along the 4-Hugoniot curve to
reach $V_{b}$. Therefore, $h_{4}\neq 0$.

Next, we present an essential estimate to bound $\left|h_{4}\right|$ more
precisely in terms of $\left|h_{4}\right|$. To that end, define the signed
length of $(U_{1}-U_{b})|_{(u,v)}$ and $(V_{b}-U_{3})|_{(u,v)}$ by $d_{1}$ and
$d_{4}$ on the $(u,v)$–plane:

$\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\|(U_{1}-U_{b})|_{(u,v)}\|&\mbox{ if
$h_{1}>0$,}\\\\[5.69054pt] -\|(U_{1}-U_{b})|_{(u,v)}\|&\mbox{ if
$h_{1}<0$,}\\\ \end{array}\right.$

and

$\displaystyle d_{4}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\|(V_{b}-U_{3})|_{(u,v)}\|&\mbox{ if
$h_{4}>0$,}\\\\[5.69054pt] -\|(V_{b}-U_{3})|_{(u,v)}\|&\mbox{ if
$h_{4}<0$.}\\\ \end{array}\right.$

Secondly, we note that

$|\lambda_{2}-\dot{z}_{b}|=\mathcal{O}(1)|d_{1}|=\mathcal{O}(1)|h_{1}(b)|.$

Moreover, since
$\lambda_{2}=\frac{v(U_{1})}{u(U_{1})}=\frac{v(U_{2})}{u(U_{2})}=\lambda_{3}$,
we can similarly conclude

$|\lambda_{3}-\dot{z}_{b}|=\mathcal{O}(1)|d_{1}|=\mathcal{O}(1)|h_{1}(b)|.$

Using the following projections on the $(u,v)$–plane,

$\textbf{r}_{1}|_{u}=-\textbf{r}_{4}|_{u},\quad\textbf{r}_{1}|_{v}=\textbf{r}_{4}|_{v},$
$\textbf{r}_{2}=\textbf{r}_{2}|_{(u,v)},\quad\textbf{r}_{3}|_{(u,v)}=0.$

Thirdly, we note that

$-d_{4}=\mathcal{O}(1)h_{2}(b)(\lambda_{2}-\dot{z}_{b})+\tilde{d},$

where $\tilde{d}\cos\varphi_{1}=d_{1}\cos\varphi_{2}$, $\varphi_{1}$ denotes
the angle between $(1,\dot{z}_{b})$ and $\textbf{r}_{4}|_{(u,v)}$,
$\varphi_{2}$ denotes the angle between $\textbf{r}_{1}|_{(u,v)}$ and
$(1,\dot{z}_{b})$, $\varphi_{1}=\varphi_{2}+2\alpha$ for $\alpha={\rm
arctan}(\dot{z}_{b})$, and

$\displaystyle\tilde{d}$ $\displaystyle=$ $\displaystyle
d_{1}\frac{\cos\varphi_{2}}{\cos\varphi_{1}}=d_{1}\frac{\cos(\varphi_{1}-2\alpha)}{\cos\varphi_{1}}=d_{1}\frac{\cos\varphi_{1}\cos(2\alpha)+\sin\varphi_{1}\sin(2\alpha)}{\cos\varphi_{1}}$
$\displaystyle=$ $\displaystyle
d_{1}\big{(}\text{cos}(2\alpha)+\mathcal{O}(1)\text{sin}(2\alpha)\big{)}=d_{1}\big{(}1+\mathcal{O}(1)\alpha\big{)}=d_{1}\big{(}1+\mathcal{O}(1)\dot{z}_{b}\big{)}.$

Hence, there holds

$-d_{4}=\mathcal{O}(1)h_{2}(b)(\lambda_{2}-\dot{z}_{b})+d_{1}\big{(}1+\mathcal{O}(1)\dot{z}_{b}\big{)}.$

At $U_{-}$, there also holds $\textbf{r}_{1}|_{(u,p,\rho)}$ =
-$\textbf{r}_{4}|_{(u,p,\rho)}$ and $\textbf{r}_{1}|_{v}=\textbf{r}_{4}|_{v}$,
which implies

$\frac{d_{1}}{h_{1}}=\frac{d_{4}}{h_{4}}.$

Hence we note the following key estimate:

$-h_{4}(b)=\mathcal{O}(1)h_{2}(b)(\lambda_{2}-\dot{z}_{b})+h_{1}(b)\big{(}1+\mathcal{O}(1)\dot{z}_{b}\big{)}.$
(5.15)

Estimate (5.15) now implies

$\displaystyle|h_{4}(b)|$ $\displaystyle\leq$
$\displaystyle|h_{1}(b)|+\mathcal{O}(1)|h_{2}(b)||\lambda_{2}-\dot{z}_{b}|+|h_{1}(b)|\mathcal{O}(1)|\dot{z}_{b}|$
$\displaystyle\leq$
$\displaystyle|h_{1}(b)|+\mathcal{O}(1)\big{(}|h_{2}(b)|+|\dot{z}_{b}|\big{)}|h_{1}(b)|,$

yielding

$|h_{1}(b)|=\tilde{\mathcal{O}}(1)|h_{4}(b)|\hskip 17.07164pt\text{with
}\frac{1}{2}<\bar{\mathcal{O}}(1)<\frac{3}{2},$

given that $|h_{2}(b)|+|\dot{z}_{b}|$ is always small enough, and this is
guaranteed by the sufficiently small total variation of the initial
perturbation $\widetilde{U_{0}}$ and the perturbation of the boundary. $\Box$

We note that the requirement
$\frac{\bar{v}}{\bar{u}}=\frac{\hat{v}}{\hat{u}}=\dot{z}_{b}$ in Proposition
5.2 is just the boundary condition (1.8) because $\dot{z}_{b}$ here is the
slope of the Lipschtiz wall.

Applying Proposition 5.2 now yields

$\displaystyle E_{b,1}$ $\displaystyle=$
$\displaystyle|q_{1}(b)|W_{1}(b)(\dot{z}_{b}+\lambda_{1})$ $\displaystyle=$
$\displaystyle-4B\kappa_{1}w_{1}^{b}|h_{1}(b)|\,|\lambda_{1}|+\mathcal{O}(1)|h_{1}(b)|$
$\displaystyle=$
$\displaystyle-4B\kappa_{1}w_{1}^{b}|h_{1}(b)||\lambda_{1}|+\mathcal{O}(1)|h_{4}(b)|,$
$\displaystyle E_{b,j}$ $\displaystyle=$
$\displaystyle|q_{j}(b)|W_{j}(b)(\dot{z}_{b}+\lambda_{j})=|q_{j}(b)|W_{j}(b)(\dot{z}_{b}-\lambda_{j})+2\lambda_{j}|q_{j}(b)|W_{j}(b)$
$\displaystyle=$
$\displaystyle\mathcal{O}(1)w^{b}_{j}|h_{j}(b)|(\dot{z}_{b}-\lambda_{j})+\mathcal{O}(1)\lambda_{j}w^{b}_{j}|h_{j}(b)|$
$\displaystyle=$
$\displaystyle\mathcal{O}(1)h_{1}(b)=\mathcal{O}(1)h_{4}(b),\hskip
17.07164ptj=2,3,$ $\displaystyle E_{b,4}$ $\displaystyle=$
$\displaystyle|q_{4}(b)|W_{4}(b)(\dot{z}_{b}+\lambda_{4})$ $\displaystyle=$
$\displaystyle
4B\kappa_{1}w^{b}_{4}|h_{4}(b)||\lambda_{1}|+\mathcal{O}(1)|h_{4}(b)|$
$\displaystyle\leq$ $\displaystyle
4B\kappa_{1}|\lambda_{1}|w^{b}_{4}\big{(}|h_{1}(b)|+\mathcal{O}(1)|h_{2}(b)||\lambda_{2}-\dot{z}_{b}|+\mathcal{O}(1)|h_{1}(b)||\dot{z}_{b}|\big{)}+\mathcal{O}(1)|h_{4}(b)|.$

Using Lemma 5.1, we can choose $w^{b}_{1}$ and $w^{b}_{4}$ such that

$w^{b}_{4}<w^{b}_{1}.$

Then, with the total variation of the incoming flow perturbation and the
boundary perturbation small enough and $\kappa_{1}$ large enough, one has

$\displaystyle\sum_{j=1}^{4}E_{b,j}$ $\displaystyle=$ $\displaystyle
4B\kappa_{1}(w^{b}_{4}-w^{b}_{1})|h_{1}(b)||\lambda_{1}|$
$\displaystyle+\mathcal{O}(1)B\kappa_{1}|\lambda_{1}|h^{b}_{4}\big{(}|h_{2}(b)|+|\dot{z}_{b}|\big{)}|h_{1}(b)|+\mathcal{O}(1)|h_{4}(b)|$
$\displaystyle\leq$
$\displaystyle\tilde{\mathcal{O}}(1)4B\kappa_{1}(w^{b}_{4}-w^{b}_{1})|h_{4}(b)|\,|\lambda_{1}|$
$\displaystyle+\mathcal{O}(1)B\kappa_{1}|\lambda_{1}|w^{b}_{4}\big{(}|h_{2}(b)|+|\dot{z}_{b}|\big{)}|h_{1}(b)|+\mathcal{O}(1)|h_{4}(b)|\leq
0,$

provided that $|h_{2}(b)|+|\dot{z}_{b}|$ is sufficiently small. This is
guaranteed since the total variation of the incoming flow perturbation and the
boundary perturbation are sufficiently small.

##  6\. Existence of a Semigroup of Solutions

As a corollary of the essential estimates in Sections 3–5, we can now
establish the existence of the semigroup $\mathscr{S}$ generated by the wave-
front tracking method, as well as the Lipschitz continuity of $\mathscr{S}$.

Proposition 6.1. _Suppose that
$TV(\widetilde{U_{0}}(\cdot))+TV(g^{\prime}(\cdot))$ is small enough. Then the
map_

$(\overline{U}(\cdot),x)\mapsto
U^{\vartheta}(x,\cdot)\mathrel{\mathop{:}}=\mathscr{S}^{\vartheta}_{x}(\overline{U}(\cdot))$

_produced by the wave-front tracking algorithm is a uniformly Lipschitz
continuous semigroup satisfying the properties_ :

  * (i)

$\mathscr{S}^{\vartheta}_{0}\overline{U}=\overline{U},\quad\mathscr{S}^{\vartheta}_{x_{1}}\mathscr{S}^{\vartheta}_{x_{2}}\overline{U}=\mathscr{S}^{\vartheta}_{x_{1}+x_{2}}\overline{U}$,
for all $x_{1},x_{2}\geq 0$;

  * (ii)

$\lVert\mathscr{S}^{\vartheta}_{x}\overline{U}-\mathscr{S}^{\vartheta}_{x}\overline{V}\rVert_{L^{1}}\leq
C\lVert\overline{U}-\overline{V}\rVert_{L^{1}}+C\vartheta x$, for all $x\geq
0$.

Proof. Since $\mathscr{S}^{\vartheta}$ is generated by the wave-front tracking
algorithm, property (i) is immediate. Next, property (ii) is proved as
follows. Take a pair of front tracking $\vartheta$-approximate solutions
$U^{\vartheta}$ and $V^{\vartheta}$ of (1.1) and (1.7)–(1.8) with
$\overline{U}(\cdot)$ and $\overline{V}(\cdot)$ as the initial data,
respectively. Using (4.11) and (4.21), at any $x\geq 0$, we have

$\displaystyle\lVert U^{\vartheta}(x)-V^{\vartheta}(x)\rVert_{L^{1}}$
$\displaystyle\leq$ $\displaystyle{C}\Phi(U^{\vartheta}(x),V^{\vartheta}(x))$
$\displaystyle\leq$
$\displaystyle{C}\Phi(U^{\vartheta}(0),V^{\vartheta}(0))+C\nu x$
$\displaystyle\leq$ $\displaystyle
C\lVert\overline{U}-\overline{V}\rVert_{L^{1}}+C\vartheta x.$

Hence, the $\vartheta$-semigroup is Lipschitz continuous.

Definition 6.1. For a given $\nu_{0}>0$, we define the domain:

$\mathcal{D}=cl\begin{cases}\mbox{the set consisting of points
}U:\mathbb{R}\mapsto\mathbb{R}^{4}\\\ \mbox{such that there exists one point
}y^{i}\in\mathbb{R}\mbox{ and}\\\
\tilde{U}(y)=\begin{cases}U_{-},\qquad&g(x)\leq y\leq y^{i},\\\
U_{+},\qquad&y^{i}<y,\end{cases}\\\ \mbox{so that }U-\tilde{U}\in
L^{1}(\mathbb{R};\mathbb{R}^{4})\mbox{ and }{\rm
TV}(U-\tilde{U})\leq\nu_{0}.\end{cases}$

Remark 6.1. Given a solution $U(x,y)$ to the initial-boundary value problem of
(1.1) and (1.7)–(1.8), we note that, if
$U^{x}(y)\mathrel{\mathop{:}}=U(x,y)\in\mathcal{D}$ at any fixed $x\geq 0$,
then $y^{i}>g(0)=0$ at $x=0$ and $y^{i}>g(x)$ for $x>0$ as a strong vortex
sheet/entropy wave is present.

The semigroup $\mathscr{S}$ generated by the wave-front tracking algorithm is
provided by the next theorem.

Theorem 6.1. _Suppose that
$TV(\widetilde{U_{0}}(\cdot))+TV(g^{\prime}(\cdot))$ is small enough. Then, in
the $L^{1}$–norm, $\mathscr{S}^{\vartheta}$ produced by the wave-front
tracking algorithm is a Cauchy sequence. Denote this unique limit by
$\mathscr{S}$ such that, for any $x>0$,
$\mathscr{S}_{x}(\overline{U})=\lim_{\vartheta\to
0}\mathscr{S}^{\vartheta}_{x}(\overline{U})$. Then the map
$\mathscr{S}:[0,\infty)\times\mathcal{D}\mapsto\mathcal{D}$ is a uniformly
Lipschtiz semigroup in $L^{1}$._ _In particular, the entropy solution to the
initial-boundary problem ( 1.1) and (1.7)–(1.8) constructed by the wave-front
tracking algorithm is unique and $L^{1}$ stable_.

Based on the essential estimates in Sections 3–5, the proof of Theorem 6.1 can
be shown in the same way as the argument given in [7]. Also see Chen-Li [10].

##  7\. Uniqueness of entropy solutions in the viscosity class

In this section, as an immediate consequence of the estimates obtained in
Sections 4–6, we find that the semigroup $\mathscr{S}$ produced by the wave-
front tracking method is the only standard Riemann semigroup (SRS) in the
sense of Definition 7.1 given below. In other words, the semigroup defined by
the wave-front tracking method is the canonical trajectory of the standard
Riemann semigroup (SRS). This yields the uniqueness of entropy solutions in a
broader class of viscosity solutions as introduced by Bressan in [5].
Furthermore, it coincides with the semigroup trajectory generated by the wave-
front tracking method.

Definition 7.1. Problem (1.1) and (1.7)–(1.8) is said to have a standard
Riemann semigroup (SRS) if, for some small $\nu_{0}$, there exist a continuous
mapping $\mathscr{R}:[0,\infty)\times\mathcal{D}\mapsto\mathcal{D}$ and a
constant $L$ satisfying the following properties:

  * (i)

(Semigroup property):
$\mathscr{R}_{0}\overline{U}=\overline{U},\quad\mathscr{R}_{x_{1}}\mathscr{R}_{x_{2}}\overline{U}=\mathscr{R}_{x_{1}+x_{2}}\overline{U}$;

  * (ii)

(Lipschitz continuity):
$\lVert\mathscr{R}_{x}\overline{U}-\mathscr{R}_{x}\overline{V}\rVert_{L^{1}}\leq
L\lVert\overline{U}-\overline{V}\rVert_{L^{1}}$;

  * (iii)

(Consistency with the Riemann solver): _Given piecewise constant initial data
$\overline{U}\in\mathcal{D}$, then, for all $x\in[0,\nu_{0}]$, the function
$U(x,\cdot)=\mathscr{S}_{x}\overline{U}$ coincides with the solution of (1.1)
and (1.7)–(1.8) obtained by piecing together the standard Riemann solutions
and the lateral Riemann solutions._

Following the argument in [5], we employ the estimates obtained in Sections
4–6 to conclude

Theorem 7.1. _Suppose that problem ( 1.1) and (1.7)–(1.8) has a standard
Riemann semigroup $\mathscr{R}:[0,\infty)\times\mathcal{D}\mapsto\mathcal{D}$.
Consider the semigroup $\mathscr{S}$ produced by the wave-front tracking
method, that is, $\mathscr{S}_{x}(\overline{U})=\lim_{\vartheta\rightarrow
0}\mathscr{S}^{\vartheta}_{x}(\overline{U})$. Assume
$\overline{U}\in\mathcal{D}$. Then, for all $x>0$,
$\mathscr{R}_{x}\overline{U}=\mathscr{S}_{x}\overline{U}$_. _Furthermore, a
continuous map $U:[0,X]\mapsto\mathcal{D}$ is a viscosity solution of problem
(1.1) and (1.7)–(1.8) defined in [5] if and only if_

$U(x,\cdot)=\mathscr{R}_{x}\overline{U}\hskip 17.07164pt\textit{for any
}x\in[0,T].$ (7.1)

_In particular, a continuous map $U:[0,X]\mapsto\mathcal{D}$ is a viscosity
solution if and only if_

$U(x,\cdot)=\mathscr{S}_{x}\overline{U}\hskip 17.07164pt\textit{for any
}x\in[0,T].$ (7.2)

The proof here follows a similar argument to the one presented in [5]. The
only difference is that there is a strong vortex sheets/entropy waves in our
problem. Nonetheless, one can proceed with the proof by considering the
convergence of the wave-front tracking method which is shown in Section 3.

Remark 7.1. In the simpler cases of the isentropic or isothermal Euler flow
(1.5), as well as the potential flow, as far as the $L^{1}$–stability problem
is of concern, we realize the same results as for the full Euler system (1.1).

Acknowledgements: The research of Gui-Qiang Chen was supported in part by the
National Science Foundation under Grants DMS-0935967 and DMS-0807551, the UK
EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE
(EP/E035027/1), the NSFC under a joint project Grant 10728101, and the Royal
Society–Wolfson Research Merit Award (UK). The research of Vaibhav Kukreja was
supported in part by the National Science Foundation under Grants DMS-0935967
and DMS-0807551, the UK EPSRC Science and Innovation Award to the Oxford
Centre for Nonlinear PDE (EP/E035027/1).

## References

  * [1]
  * [2] P. Baiti and K. Jenssen, On the front-tracking algorithm, J. Math. Anal. Appl. 217 (1998), 395–404.
  * [3] S. Biachini and A. Bressan, Vanishing viscosity of nonlinear hyperbolic systems, Ann. of Math. 161 (1) (2005), 223–342.
  * [4] A. Bressan, Global solutions of systems of conservation laws by wave-front tracking, J. Math. Anal. Appl. 170 (1992), 414–432.
  * [5] A. Bressan, The unique limit of the Glimm scheme, Arch. Ration. Mech. Anal. 130 (1995), 205–230.
  * [6] A. Bressan, Hyperbolic Systems of Conservations Laws: The One-Dimensional Cauchy Problem, Oxford Univ. Press, Oxford, 2000.
  * [7] A. Bressan and R. M. Colombo, The semigroup of $2\times 2$ conservation laws, Indiana Univ. Math. J. 44 (1995), 677–725.
  * [8] A. Bressan, G. Crasta, and R. M. Colombo, Well-posedness of the Cauchy problem for $n\times n$ systems of conservation laws, Mem. Amer. Math. Soc. 146 (694), 2000.
  * [9] A. Bressan, T.-P. Liu, and T. Yang, $L^{1}$ stability estimates for $n\times n$ conservation laws, Arch. Ration. Mech. Anal. 149 (1999), 1–22.
  * [10] G.-Q. Chen and T.-H. Li, Well-posedness for two-dimensional steady supersonic Euler flows past a Lipschitz wedge, J. Differential Equations, 244 (2008), 1521–1550.
  * [11] G.-Q. Chen, Y.-Q. Zhang, and D.-W. Zhu, Stability of compressible vortex sheets in steady supersonic Euler flows over Lipschitz walls, SIAM J. Math. Anal. 38 (2007), 1660–1693.
  * [12] R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948.
  * [13] A. Corli and M. Sabl$\acute{\text{e}}$-Tougeron, Stability of contact discontinuities under perturbations of bounded variation, Rend. Sem. Mat. Univ. Podova, 97 (1997), 35–60.
  * [14] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41.
  * [15] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Second ed., Springer-Verlag, Berlin, 2005.
  * [16] R. J. DiPerna, Global existence of solutions to nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 20 (1976), 187–212.
  * [17] J. Glimm, Solution in the large for nonlinear systems of conservation laws, Comm. Pure Appl. Math. 18 (1965), 697–715.
  * [18] H. Holden and N. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2002.
  * [19] Ph. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Birkh$\ddot{\text{a}}$user-Verlag, Basel, 2002.
  * [20] M. Lewicka, $L^{1}$ stability of patterns of non-interacting large shock waves, Indiana Univ. Math. J. 49 (2000), 1515–1537.
  * [21] M. Lewicka, Stability conditions for patterns of noninteracting large shock waves, SIAM J. Math. Anal. 32 (2001), 1094–1116.
  * [22] M. Lewicka and K. Trivisa, On the $L^{1}$ well posedness of systems of conservation laws near solutions containing two large shocks, J. Differential Equations, 179 (2002), 133–177.
  * [23] T.-P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135–148.
  * [24] T.-P. Liu and T. Yang, Well-posedness theory for hyperbolic conservation laws, Comm. Pure Appl. Math. 52 (1999), 1553–1586.
  * [25] M. Sabl$\acute{\text{e}}$-Tougeron, M$\acute{\text{e}}$thode de Glimm et probl$\grave{\text{e}}$me mixte, Ann. Inst. H. Poincar$\acute{\text{e}}$ Anal. Nonlin$\acute{\text{e}}$aire, 10 (1993), 423–443.