Two-Dimensional Riemann Problem with Four-Shock Interactions]Global Solutions of the Two-Dimensional

Riemann Problem with Four-Shock Interactions
for the Euler Equations for Potential Flow

Gui-Qiang G. Chen
[Gui-Qiang G. Chen]Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

Alex Cliffe
[Alex Cliffe]Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

Feimin Huang
[Feimin Huang]Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Song Liu
[Song Liu]Department of Mathematics, City University of Hong Kong, Hong Kong, China;
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Qin Wang
[Qin Wang]Department of Mathematics, Yunnan University, Kunming, 650091, China

We present a rigorous approach and related techniques to construct global solutions
of the two-dimensional (2-D) Riemann problem with four-shock interactions for
the Euler equations for potential flow.
With the introduction of three critical angles: the vacuum critical angle from the compatibility conditions,
the detachment angle, and the sonic angle, we clarify all
configurations of the Riemann solutions for the interactions of two-forward and two-backward shocks,
including the subsonic-subsonic reflection configuration that has not
emerged in previous results.
To achieve this, we first identify the three critical angles that determine the configurations,
whose existence and uniqueness follow from our
rigorous proof of the strict monotonicity of the steady detachment
and sonic angles for 2-D steady potential flow
with respect to the Mach number of the upstream state.
Then we reformulate the 2-D Riemann problem into the shock reflection-diffraction problem
with respect to a symmetric line, along with
two independent incident angles
and two sonic boundaries varying with the choice of incident angles.
With these, the problem can be further reformulated as a free boundary problem
for a second-order
quasilinear equation
of mixed elliptic-hyperbolic type.
The difficulties arise from the degenerate ellipticity of the
nonlinear equation
near the sonic boundaries, the nonlinearity of the free boundary condition,
the singularity of the solution near the corners of the domain,
and the geometric properties of the free boundary.
To solve the problem, we need to analyze
the solutions for a quasilinear degenerate elliptic equation
by the maximum principle of the mixed-boundary value problem,
the theory of the oblique derivative boundary value problem, the uniform
a priori estimates, and the iteration method.
To the best of our knowledge, this is the first rigorous result
for the 2-D Riemann problem with four-shock interactions
for the Euler equations.
The approach and techniques developed for the Riemann problem for four-wave interactions
should be useful for solving other 2-D Riemann problems
for more general Euler equations and
related nonlinear hyperbolic systems of conservation laws.

[2020]35M30, 35M12, 35R35, 35Q31, 76N10, 35L65, 35L70, 35B65, 35D30, 35J70, 35J66

§ INTRODUCTION

We are concerned with the two-dimensional (2-D) Riemann problem for the Euler equations for potential flow,
consisting of the conservation law of mass
and the Bernoulli law:

\begin{align}\label{System:2DPotentialFlowEqs}
\begin{cases}
\, \partial_{t}\rho+{\rm div}_{\mathbf{x}}(\rho \nabla_{\mathbf{x}}\Phi) = 0\,,\\[1mm]
\, \partial_{t}\Phi+\frac{1}{2}\abs{\nabla_{\mathbf{x}}\Phi}^2+h(\rho) = B\,,
\end{cases}
\end{align}

where $(t,\mathbf{x})\in(0,\infty)\times\mathbb{R}^2$, $\rho = \rho(t,\mathbf{x})$ is the density, $\Phi=\Phi(t,\mathbf{x})$ is the velocity potential
(i.e., $\nabla_{\mathbf{x}}\Phi = (u,v)$
is the velocity),
$h(\rho) \defeq \int_{1}^{\rho}\frac{p'(s)}{s} \, {\rm d} s$ is the enthalpy with the polytropic pressure law $p(\rho)=A\rho^{\gamma}$ for $\gamma\geq1$ and $A > 0$, and
the Bernoulli constant \(B\) is determined by the far field flow.
Without loss of generality, we may fix $A = \gamma^{-1}$ by scaling and
denote the sonic speed by $c(\rho)\defeq \sqrt{p'(\rho)} = \rho^{\frac{\gamma-1}{2}}$.

In this paper, we focus on the 2-D Riemann problem with four-shock interactions and corresponding Riemann initial
given by

\begin{equation}
\label{Eq:RiemannInitialData}
(\rho,\Phi)(0 , \mathbf{x}) = (\rho_i, (u_i, v_i) \cdot \mathbf{x} ) \qquad\,\, \mbox{for $\mathbf{x} \in \Lambda_{i}$}\,,
\end{equation}

for suitable constant states $(i)$
with values \(U_i \defeq (\rho_i, u_i, v_i),\)
and corresponding scale-invariant domains $\Lambda_i \subseteq \mathbb{R}^2$ for $i=1,2,3,4$.
We choose these domains to be four sectors of \(\mathbb{R}^2,\) defined by

\begin{align}\label{Eq:InitialDomain}
\begin{aligned}
&\quad \Lambda_1 \defeq \big\{\mathbf{x} \in \mathbb{R}^2 \,:\, -{\theta}_{14} < \theta < {\theta}_{12} \big\} \,,
\Lambda_2 &\defeq \big\{\mathbf{x} \in \mathbb{R}^2 \,:\, {\theta}_{12} < \theta < \pi-{\theta}_{32} \big\}\,,\\
&\quad \Lambda_3 \defeq \big\{\mathbf{x} \in \mathbb{R}^2 \,:\, \pi-{\theta}_{32} < \theta < \pi+{\theta}_{34} \big\}\,,
\Lambda_4 &\defeq \big\{\mathbf{x} \in \mathbb{R}^2 \,:\, \pi+{\theta}_{34} < \theta < 2\pi-{\theta}_{14} \big\}\,,
\end{aligned}
\end{align}

where \(\theta\) is the polar angle of point \(\mathbf{x}\in\mathbb{R}^2,\)
and the four parameters \({\theta}_{12}, {\theta}_{32}, {\theta}_{34}, {\theta}_{14} \in (0,\frac{\pi}{2})\).
The initial data and shock discontinuities are depicted in Fig. <ref>.

The Euler equations for potential flow (<ref>) are
the oldest, yet still prominent, paradigm of fluid dynamic equations for inviscid fluids,
which have been widely used in aerodynamics.
System (<ref>)
has further been extended to the full Euler equations
for more general inviscid compressible fluids when the effect of vortex sheets and the derivation of
vorticity become significant.
One of the main features of the Euler equations is the formation of shocks
in finite time,
no matter how smooth the initial data are.
The shock is one of the most fundamental nonlinear waves
in compressible fluid flow,
and the analysis of shocks dates back to the 19th century;
see e.g. [20, 27, 43, 44, 47].
In 1860, Riemann [44] first considered a special initial value problem
for the one-dimensional  Euler system, with two constant states
separated at the origin—now known as the Riemann problem.
For general 1-D strictly hyperbolic systems of conservation laws,
Lax [29] showed that the Riemann solutions are the combination of
three fundamental waves: the rarefaction waves, the shocks, and the contact discontinuities.
In 1965, Glimm [24] employed the Riemann solutions as building blocks
of the Glimm scheme to establish the global existence of solutions in BV (bounded variation)
of 1-D strictly hyperbolic systems of conservation laws
for the initial data of small BV, for which the Riemann solutions play a fundamental role.
For the uniqueness and stability of BV solutions,
we refer to [3, 4, 5, 40, 41] and the references cited therein.
For a systematic theory of 1-D hyperbolic systems of conservation laws,
we refer to [3, 19, 26, 30, 42]
and the references cited therein.

The four sectors of the Riemann initial data.

Multi-dimensional (M-D)
Riemann problems for the Euler system are vastly more
complicated; see [8, 19, 26, 30, 31]
and the references cited therein.
For the 2-D four-quadrant Riemann problem with four initial constant states
given in the four quadrants
so that each jump between two neighbouring quadrants projects exactly one planar fundamental wave,
Zhang-Zheng [49] predicted that there are a total of 16 genuinely different configurations
of the Riemann solutions for polytropic gases.
In Chang-Chen-Yang [6, 7],
it is first observed that, when two initially parallel slip lines are present,
it makes a distinguished difference whether the vorticity waves generated have the same or opposite sign,
which, along with Lax-Liu [32], leads to the classification with a total of 19 genuinely
different configurations of the Riemann solutions for the compressible Euler equations
for polytropic gases, via characteristic analysis;
also see [25, 28, 33, 45] and the references cited therein.
However, there have been few rigorous global results
for the 2-D four-quadrant Riemann problem for the Euler equations,
except for the result obtained by Li-Zheng in [36], in which the four-rarefaction wave case was considered.
On the other hand, the 2-D Riemann problem has been
solved for simplified models such as the pressure gradient system [14],
and the local structure of wave interactions in the Riemann problem has been analyzed in [34, 35, 37].
For the 2-D Riemann problem for Chaplygin gases, we refer to [17, 46] and the references cited therein.

The purpose of this paper is to solve rigorously the 2-D Riemann problem with four-shock interactions
globally for the Euler equations for potential flow (<ref>).
The 2-D Riemann initial data (<ref>) under consideration in this paper satisfy that

\begin{equation*}
\begin{cases}
\,\text{one forward shock}\;
\overrightarrow{S}\!_{1j}\;\text{is generated between states} \; (1) \;\text{and}\; (j),\\[1mm]
\,\text{one backward shock}\; \overleftarrow{S}\!_{3j} \;\text{is generated between states} \; (3)\;\text{and}\; (j),
\end{cases}
\end{equation*}

for $j=2,4$,
and that the angular bisectors at the vertices of $\Lambda_{1}$ and $\Lambda_{3}$ in Fig. <ref> coincide.
Without loss of generality, we choose $\theta_{12}=\theta_{14}$ and $\theta_{32}=\theta_{34}$,
and then the $x_1$–axis may be considered as a rigid wall.
The above Riemann problem corresponds to one of the conjectures for
the Euler equations proposed in Zhang-Zheng [49].

The problem considered has a close relation to the shock reflection-diffraction problem, whose research dates back to 1878, in which Ernst Mach revealed the complexity of
the shock reflection-diffraction configurations that include two fundamental modes:
the regular reflection-diffraction and
the Mach reflection-diffraction.
The shock reflection phenomena occur when a moving planar shock wave impinges on a rigid wall;
see Courant-Friedrichs [18] and Chen-Feldman [10].
In the recent three decades, there have been many theoretical results
on shock reflection/diffraction problems.
Indeed, Elling-Liu [22]
considered the self-similar solutions of the Prandtl-Meyer problem for supersonic potential flow
onto a solid wedge for a class of physical parameters.
Chen-Feldman [9] proved the first global existence
of solutions of the shock reflection-diffraction problem for potential flow
when the incident angles are small; later, they succeeded to establish the global existence of solutions
for the incident angles all the way up to the detachment angle in [10].
Furthermore, Chen-Feldman-Xiang [12]
proved the convexity of transonic shocks, which leads to the uniqueness
and stability of the admissible solution [13].
More recently, Bae-Chen-Feldman [2] obtained the existence and the optimal regularity of admissible
solutions of the Prandtl-Meyer reflection problem for the general case up to the detachment angle.
For the supersonic flow passing through a wing or curved wedge, see also [15, 16, 11].

For the 2-D Riemann problem with four-shock interactions under consideration,
we can reformulate the problem into a shock reflection-diffraction problem with respect to a symmetric line.
Compared with the previous results, such as the shock reflection-diffraction problem [10]
and the Prandtl-Meyer reflection problem [2],
we now have two independent incident angles in the four-shock reflection problem.
Correspondingly, we have two sonic boundaries varying with the choice of incident angles,
which leads to more complicated configurations.
After introducing the self-similar coordinates, the original problem can
be reformulated as a free boundary problem for a second-order quasilinear
partial differential equation (PDE)
of mixed elliptic-hyperbolic type.
The difficulties come from the degenerate ellipticity of the nonlinear PDE
near the sonic boundaries, the nonlinearity of the free boundary conditions,
the singularity of the solution near the corners of the domain,
and the geometric properties of the free boundary.
To solve our problem, we need to
the solutions
for a quasilinear degenerate elliptic equation
by the maximum principle of mixed-boundary value problems
and the theory of the oblique derivative boundary value
problems [23, 38, 39],
as well as the uniform estimates and the iteration method
from [2, 10, 21].

An important part of our analysis includes the determination of the
three critical angles: the vacuum critical angle, the detachment angle, and the sonic angle.
Their existence and uniqueness follow
from Lemma <ref>,
whereby we prove the strict monotonicity of the steady detachment
and sonic angles for 2-D steady potential flow with respect to
the Mach number of the upstream state.
Using these critical angles,
we classify all configurations of the solutions of the Riemann problem.
In particular, the subsonic-subsonic reflection configuration
has never emerged in the previous works.
Lemma <ref> should be
useful in the analysis
of other shock reflection-diffraction problems.

To the best of our knowledge, this is the first result on the shock reflection-diffraction
of the Riemann problem for potential flow with piecewise constant initial data
in four sectors.
Furthermore, the mathematical methods developed in solving the free boundary problem
for nonlinear PDEs such as the uniform a priori estimate on the ellipticity of
the equation,
the anisotropic scaling method, and the sophisticated construction of the iteration set
and iteration map will provide insights for the analysis of more general
nonlinear problems.
In particular, the Euler equations for potential flow are the core part of
the full Euler equations, coupled with the incompressible-type Euler equations,
in a local domain, the techniques and approaches developed for the Riemann problem for four-wave interactions
will be useful to guide the analysis of the structure of Riemann solutions, shock reflection/diffraction problems,
and numerical simulations of the full Euler equations.

The organisation of this paper is as follows:
In Section <ref>,
we present the mathematical formulation of the Riemann problem with four-shock interactions
and the definition of admissible solutions of the free boundary problem,
and state three main results.
In Section <ref>,
we obtain the directional monotonicity, the strict ellipticity, and the uniform weighted
Hölder estimates of admissible solutions.
In Section <ref>,
we prove the existence of admissible solutions by constructing a suitable iteration set and iteration map via
applying the Leray-Schauder degree theory.
Finally, in Section <ref>,
we give the proof of the optimal regularity of admissible solutions,
as well as the proof of the convexity of free boundaries and transonic shocks.
In Appendix <ref>, we give the details for some calculations
involving the potential flow equation
and, in Appendix <ref>,
we present some known results that are needed throughout the proofs.

§ THE RIEMANN PROBLEM AND MAIN THEOREMS FOR FOUR-SHOCK INTERACTIONS

In this section, we first present the mathematical formulation of the Riemann problem
with four-shock interactions
as a free boundary problem, along with the definition of admissible solutions of the free boundary problem,
and then state three main theorems.

§.§ Mathematical formulation

The Riemann problem (<ref>)–(<ref>) is invariant
under the self-similar scaling:

\begin{align*}
(t,\mathbf{x}) \mapsto (\lambda t, \lambda \mathbf{x})\,,
\quad (\rho , \Phi)(t,\mathbf{x}) \mapsto (\rho,
\frac{1}{\lambda} \Phi)(\lambda t, \lambda \mathbf{x}) \qquad\;\; \mbox{for any $\lambda>0$}\,.
\end{align*}

This leads us to consider the self-similar solution $(\rho,\phi)(\bm{\xi})\defeq(\rho,\frac{\Phi}{t})(t, \mathbf{x})$
in the self-similar coordinates $\bm{\xi} = (\xi,\eta) \defeq
\frac{\mathbf{x}}{t} \in\mathbb{R}^2$.
Moreover, by introducing the pseudo-potential function $\varphi(\bm{\xi})\defeq\phi(\bm{\xi})-\frac12|\bm{\xi}|^2$,
we obtain the pseudo-steady potential flow equation for $\varphi(\bm{\xi})$ in the form:

\begin{eqnarray}
\label{Eq4PseudoPoten}
\mbox{div}(\rho(|D\varphi|,\varphi)D\varphi)+2\rho(|D\varphi|,\varphi) = 0\,,
\end{eqnarray}

where \({\rm div}\) and \(D\) represent the divergence and gradient operators with respect to the self-similar
coordinates $\bm{\xi}\in\mathbb{R}^2$, and the density is determined from the Bernoulli law as

\begin{eqnarray}\label{Relat4Rho}
\rho(|D\varphi|,\varphi) =
\begin{cases}
\big(1+(\gamma-1)(B-\frac12|D\varphi|^2-\varphi)\big)^{\frac{1}{\gamma-1}}
&\quad \text{if} \; \gamma>1\,,\\
\,\text{exp}(B-\frac12|D\varphi|^2-\varphi) &\quad \text{if} \; \gamma=1\,.
\end{cases} \end{eqnarray}

Equation (<ref>) with (<ref>) is a second-order nonlinear equation of mixed elliptic-hyperbolic type.
It is elliptic if and only if $|D\varphi|<c(|D\varphi|,\varphi)$ (pseudo-subsonic),
while it is hyperbolic if and only if $|D\varphi|>c(|D\varphi|,\varphi)$ (pseudo-supersonic).

We seek a weak solution for the pseudo-steady potential flow equation (<ref>)–(<ref>).

A function  $\varphi\in W^{1,\infty}_{{\rm loc}}(\mathbb{R}^2)$ is called a weak solution
of equation (<ref>)–(<ref>) in $\mathbb{R}^2$
if $\varphi$ satisfies the following properties:

$B-\frac12|D\varphi|^2-\varphi \geq  h(0^+) \quad \text{a.e.~in } \mathbb{R}^2 ;$

$\int_{\mathbb{R}^2}\big(\rho(|D\varphi|,\varphi)D\varphi\cdot D\zeta-2\rho(|D\varphi|,\varphi)\zeta\big)
\, {\rm d}\bm{\xi}=0\,$ for every $\zeta\in C^{\infty}_{\rm {c}}(\mathbb{R}^2)$.

Let $S$ be a $C^1$–curve across which $|D\varphi|$ is discontinuous.
Then $\varphi$ is a weak solution if and only if $\varphi$ satisfies the Rankine–Hugoniot conditions on $S$:

\begin{eqnarray}\label{RHCondi}
\end{eqnarray}

where $[F]_{S}$ represents the jump of a quantity $F$ across $S$,
and $\bm{\nu}_{S}$ is any unit normal vector on $S$.
By Definition <ref>(<ref>)
and (<ref>),
\(D\varphi \cdot \bm{\nu}_{S}\) is either positive or negative across the discontinuity \(S\).
We say that \(\bm{\nu}_{S}\) points from upstream to downstream
if \(D\varphi \cdot \bm{\nu}_{S} > 0\),
and from downstream to upstream if \(D\varphi \cdot \bm{\nu}_{S} < 0\).
The discontinuity $S$ is called a shock if it further satisfies
the physical entropy condition: the corresponding density $\rho(|D\varphi|,\varphi)$
increases across $S$ from upstream to downstream,
or equivalently, \(|D\varphi \cdot \bm{\nu}_S|\) decreases across \(S\)
from upstream to downstream.

In the self-similar coordinates $\bm{\xi}=(\xi,\eta)$, the initial condition (<ref>)
becomes the following asymptotic boundary condition at infinity:
For any \(\theta \in [0,2\pi)\) and \(i = 1,2,3,4\),

\begin{align*}
\lim_{r \to \infty} \norm{\varphi - \varphi_i}_{C^{0,1}(R_\theta \setminus B_r(\bm{0}))} = 0\,,
\end{align*}

whenever ray \(R_\theta \defeq \{ r(\cos{\theta},\sin{\theta})\,:\,r > 0 \}\) is contained
in sector \(\Lambda_i\) as defined in (<ref>),
where the pseudo-potential function \(\varphi_i\) corresponding to
the uniform state \( (i)\) is given by

\begin{align} \label{4States}
\varphi_i (\bm{\xi}) \defeq -\frac12 \abs{\bm{\xi}}^2 + (u_i,v_i) \cdot \bm{\xi} + k_i \,,
\end{align}

and constant \(k_i\) is uniquely determined to satisfy (<ref>)
with \((\rho, \varphi) = (\rho_i,\varphi_i)\) for $i=1,2,3,4$.
We may simply fix \(k_2 = 0\) in (<ref>) since the Bernoulli constant can be shifted to \(B - k_2\).

Since system (<ref>) is of Galilean
it is convenient to normalize the initial data to satisfy \(u_2 = v_1 = 0\).
From (<ref>) and (<ref>) with \(k_2 = 0\), the Bernoulli constant \(B\) is then fixed as

\begin{align} \label{eq:bernoulli-constant-def}
B = \frac12 v_2^2 + h(\rho_2)\,.
\end{align}

In this paper, we assume that the initial data (<ref>) are
chosen such that,
in the far field region, any two neighboring states are connected by exactly one planar shock discontinuity.
Furthermore, we restrict our attention to the case:

\begin{align} \label{eq:initial-data-entropy-condition}
\max\{ \rho_1,\rho_3 \} < \min\{ \rho_2, \rho_4\}\,,
\end{align}

so that the initial data consist
precisely of two forward shocks
$\{\overrightarrow{S}\!_{12}, \overrightarrow{S}\!_{14}\}$
and two backward shocks $\{\overleftarrow{S}\!_{32}, \overleftarrow{S}\!_{34}\}$,
which travel away from the origin in pairs; see
Case 5.2 in [6].
Moreover, to facilitate our analysis, we focus
on the symmetric case:

\begin{align} \label{eq:symmetry-assumption}
\theta_{12}=\theta_{14}\eqdef \theta_1, \quad\,
\theta_{32}=\theta_{34}\eqdef \theta_2\, \qquad\, \text{with $\theta_1,\theta_2 \in (0,\tfrac{\pi}{2})$}\,.
\end{align}

We refer to parameters \(\btheta \defeq (\theta_1,\theta_2)\) as the incident angles of the shock discontinuities;
see Fig. <ref>.

Four-shock interaction with symmetric incident shocks.

§.§.§ Compatibility conditions

It is clear that the initial data and parameters \(\btheta\) cannot be specified arbitrarily.
In the following lemma, we give the necessary and sufficient conditions for
the initial data \(U_i = (\rho_i,u_i,v_i)\) and parameters \(\btheta\) to generate two forward
shocks $\{\overrightarrow{S}\!_{12}, \overrightarrow{S}\!_{14}\}$
and two backward shocks $\{\overleftarrow{S}\!_{32}, \overleftarrow{S}\!_{34}\}$.

Fix $\gamma\geq1$ and $\rho_2>0$.
There exist both a constant \(\vmin \in [-\infty,0)\) depending on $(\gamma,\rho_2)$
and a critical angle \(\theta^{\rm cr} \in (0,\frac{\pi}{2}]\) depending on $(\gamma,\rho_2,v_2)$
such that, whenever $v_2\in (\vmin,0)$ and $\btheta\in (0,\theta^{\rm cr})^2,$
there exists a unique choice of constant states $U_i=(\rho_i,u_i,v_i),$ for $i=1,3,4,$
satisfying the Rankine–Hugoniot conditions (<ref>) between any two neighbouring states
and the entropy condition (<ref>)
so that two forward shocks $\{\overrightarrow{S}\!_{12}, \overrightarrow{S}\!_{14}\}$
and two backward shocks $\{\overleftarrow{S}\!_{32}, \overleftarrow{S}\!_{34}\}$
are generated.

Throughout the proof, we take \(i =1,3\), and \(j = 2,4\).
In the self-similar coordinates, we denote the shock discontinuity line
between state $(i)$ and state $(j)$ by

\begin{equation} \label{eq:incident-shock-sij}
S_{ij}\defeq \{\bm{\xi} \in \mathbb{R}^2 \,:\, \varphi_i(\bm{\xi})=\varphi_j(\bm{\xi})\}
= \big\{\bm{\xi}\in\mathbb{R}^2 \,:\, \eta=(-1)^{\frac{i+j+1}{2}}\xi\tan\theta_{ij}+a_{ij}
\big\}\,,
\end{equation}

where the expressions of $\varphi_{i}$ and $\varphi_{j}$ are given in (<ref>),
$\theta_{ij}\in(0,\frac{\pi}{2})$ is the angle between the $\xi$–axis and the shock discontinuity
line $S_{ij}$, and $a_{ij}$ is a constant determined below.
We fix the unit normal vector on \(S_{ij}\) to be

\begin{equation*}
\bm{\nu}_{ij}=(
(-1)^{\frac{i+1}{2}} \sin\theta_{ij},(-1)^{\frac{j+2}{2}}\cos\theta_{ij}
\end{equation*}

which points towards the downstream state according to the entropy
condition (<ref>), so that

\begin{align} \label{eq:incident-shock-entropy-condition}
D\varphi_i(\bm{\xi}) \cdot \bm{\nu}_{ij} > D\varphi_j(\bm{\xi}) \cdot \bm{\nu}_{ij} > 0\,
\qquad\; \text{for $\bm{\xi} \in S_{ij}$}\,.
\end{align}

A straightforward calculation by using (<ref>)–(<ref>)  and (<ref>)–(<ref>) gives

\begin{equation}\label{Eq:RelationForUij}
(u_i,v_i)-(u_j,v_j)=\ell_{ij}\bm{\nu}_{ij}\,, \quad
\Big(v_i -u_i\tan\theta_{ij} +  \frac{(-1)^{\frac{i-1}{2}}\rho_j\ell_{ij}}{(\rho_i-\rho_j)\cos\theta_{ij}} \Big)\,,
\end{equation}

where  $\ell_{ij}\defeq\ell(\rho_i,\rho_j)$ is defined as

\begin{equation} \label{eq:sonic-centre-relations}
\ell(\rho_i,\rho_j) \defeq  \sqrt{ \frac{2(\rho_i-\rho_j)(h(\rho_i) - h(\rho_j))}{\rho_i+\rho_j}}\,.
\end{equation}

It is direct to check that \(\ell\) is symmetric and satisfies the strict monotonicity: For any \(\bar{\rho} > 0\),

\begin{align} \label{eq:monotonicity-of-ell}
\rho \mapsto \ell(\bar{\rho}, \rho) \;\; \text{is
strictly increasing on $\rho \in (\bar{\rho},\infty)$ and
strictly decreasing on $\rho \in (0, \bar{\rho})$}\,.
\end{align}

From (<ref>) and (<ref>), we deduce that

\begin{equation*}
\end{equation*}

from which $\rho_4 = \rho_2$ must hold by virtue of the entropy condition (<ref>) and (<ref>).
Thus, we obtain the following relations:

\begin{equation*}\label{CompatiCondi}
\ell_{12}\cos{\theta}_{1} = \ell_{32}\cos{\theta}_{2} \,,\qquad
\left\{
\begin{alignedat}{3}
\vphantom{f^2}{}
u_2 &= u_4 &&= u_1+\ell_{12}\sin{\theta}_{1} &&= u_3-\ell_{32}\sin{\theta}_{2}\,, \\
v_1 &= v_3 &&= v_2+\ell_{12}\cos{\theta}_{1} &&= v_4-\ell_{32}\cos{\theta}_{2}\,.
\end{alignedat}
\right.
\end{equation*}

Recall that $u_2 = v_1 = 0$ are fixed, so that

\begin{align} \label{eq:states-1-4}
\quad U_1 = (\rho_1,-\ell_{12}\sin\theta_1,0)\,, \,\,\,
U_2 = (\rho_2, 0 , v_2)\,, \,\,\,
U_3 = (\rho_3,\ell_{32}\sin\theta_2,0)\,,
\,\,\,
U_4 = (\rho_2,0,-v_2)\,,
\end{align}

whenever $\rho_1,\rho_3\in(0,\rho_2)$ satisfy

\begin{equation}\label{CompatiCondiGTI}
\ell_{12}\cos{\theta}_{1} = \ell_{32} \cos{\theta}_{2} = -v_2\,.
\end{equation}

We refer to (<ref>) as the compatibility conditions for the initial data.
We define

\begin{align} \label{Eq:DefThetaVac}
\vmin \defeq -\ell(0^+,\rho_2) \in [-\infty,0)\,, \qquad
\theta^{\rm cr} \defeq \arccos{\big(\frac{-v_2}{\ell(0^+,\rho_2)} \big)} \in (0, \tfrac{\pi}{2}]\,.
\end{align}

Then, by (<ref>), the necessary and sufficient conditions for the existence
of  $\rho_1$ and $\rho_3$ uniquely solving (<ref>) subject to (<ref>) are

\begin{align*}
v_2 \in (\vmin,0)\,, \qquad
\btheta \in (0,\theta^{\rm cr})^2\,.
\end{align*}

The constant states $U_i$, $i=1,3,4$, are then uniquely determined
by (<ref>)–(<ref>)
depending only on \((\gamma, \rho_2, v_2, \btheta)\).

We call $\theta^{\rm cr}$ the vacuum critical angle, because \(\rho_1 \to 0^+\) as \(\theta_1 \to \theta^{{\rm cr}-}\)
and \(\rho_3 \to 0^+\) as \(\theta_2 \to \theta^{{\rm cr}-}\), according to (<ref>).
Note that, in potential flow, the incident shock with vacuum upstream state
is mathematically consistent, which is in contrast to the non-potential Euler flow for which the maximal density ratio across a shock is bounded.

It follows from Lemma <ref> that $\rho_2 = \rho_4$.
Without loss of generality, we fix $\rho_2 = \rho_4 = 1$ via the scale-invariance of
equation (<ref>) as follows

\begin{equation*}
\bm{\xi} \mapsto c_{2} \bm{\xi}\,, \qquad
(\rho, \varphi, \rho_2, v_2) \mapsto \big(\frac{\rho}{\rho_{2}}, \frac{\varphi}{c_{2}^{2}}, 1, \frac{v_2}{c_2} \big)\,.
\end{equation*}

Then the entropy condition (<ref>)
becomes $\max\{\rho_1,\rho_3\} < 1$, and  \(v_{\min}= -\ell(0^+,1)\) depends only on \(\gamma \geq 1\).
For simplicity, we occasionally use the abbreviation $\ell(\cdot) \defeq \ell(\cdot,1)$.

§.§.§ The asymptotic boundary value problem in the upper half-plane

Equation (<ref>) and the initial data in (<ref>)
given by (<ref>) are invariant under reflection with respect to the \(x_1\)–axis.
Thus, we look for solutions of (<ref>)–(<ref>)
in the symmetric form:

\begin{equation*}
\Phi(t,x_1,x_2) = \Phi(t,x_1,-x_2)
\qquad\; \text{for all $(t,x_1,x_2) \in (0,\infty) \times \mathbb{R}^2$}\,,
\end{equation*}

which is equivalent to

\begin{align*}
\varphi(\xi,\eta) = \varphi(\xi, -\eta)\qquad \text{for all $(\xi,\eta) \in \mathbb{R}^2$}\,.
\end{align*}

For this reason, it suffices to consider the restriction of solutions to the upper half-plane

\begin{align*}
\mathbb{R}^2_+ \defeq \big\{ \bm{\xi} \in \mathbb{R}^2 \,:\, \eta > 0\big\}\,,
\end{align*}

so long as we impose the slip boundary condition

\begin{equation*} \label{eq:slip-boundary-condition-varphi-2}
D\varphi \cdot (0,1) = 0 \qquad\; \text{on $L_{\rm sym} \defeq \big\{\bm{\xi}\in\mathbb{R}^2 \,:\, \eta=0\big\}$}\,.
\end{equation*}

This condition means that \(L_{\rm sym}\) can be regarded as a solid wall.

Hereafter, we use symbols $S_{12}$ and $S_{32}$ to represent only the closure of the rays
of the incident shocks ${S}_{12}$ and ${S}_{32}$ lying in the upper half-plane.
We denote by \(P_0^1= (\xi^{P_0^1}, \eta^{P_0^1})\) the point of intersection
between \(S_{12}\) and \(L_{\rm sym}\),
and by \(P_0^2 = (\xi^{P_0^2},\eta^{P_0^2})\) the point of intersection
between \(S_{32}\) and \(L_{\rm sym}\).
We expect the incident shocks \(S_{12}\) and \(S_{32}\) to undergo shock reflection-diffraction
at points \(P_0^1\) and \(P_0^2\) respectively; see Fig. <ref>.
From (<ref>), (<ref>), and (<ref>), we have

\begin{equation}\label{P0-Pos}
\begin{aligned}
P_0^1 = ( {-}\ell(\rho_1) \sin\theta_{1}+\frac{1}{1-\rho_1}\frac{\ell(\rho_1)}{\sin{\theta}_{1}},\,0),\qquad
P_0^2 = (\ell(\rho_3) \sin\theta_{2}-\frac{1}{1-\rho_3}\frac{\ell(\rho_3)}{\sin{\theta}_{2}},\,0)\,.
\end{aligned}
\end{equation}

Using (<ref>), (<ref>)–(<ref>), and (<ref>),
the pseudo-potentials of states \((1)\)–\((3)\) are given by

\begin{equation}\label{PseudoVelocity123}
\,\,\varphi_1 = -\frac{1}{2}|\bm{\xi}|^2 + v_2 (\xi-\xi^{P_0^1}) \tan{\theta_1}, \;
\varphi_2 = -\frac{1}{2}|\bm{\xi}|^2 + v_2 \eta, \;
\varphi_3 = -\frac{1}{2}|\bm{\xi}|^2 - v_2 (\xi-\xi^{P_0^2}) \tan{\theta_2}.
\end{equation}

In particular, \(\varphi_1\) is independent of \(\theta_2\), while \(\varphi_3\) is independent of \(\theta_1\).
The following lemma concerning the pseudo-Mach number of state \((2)\)
at the reflection points \(P_0^1\) and \(P_0^2\) is fundamental to our later analysis
of the critical angles in <ref>.

Fix \(\gamma \geq 1\) and  \(v_2\in(v_{\min},0)\).
For any incident angle \(\theta_1 \in (0, \theta^{\rm cr}),\)
the pseudo-Mach number of state \((2)\) at the reflection point \(P_0^1\) is given by

\begin{align}
\label{eq:appendixMachNumberState12}
M_2^{(\theta_1)} \defeq
\abs{D\varphi_2(P_0^1)} =
\frac{\ell(\rho_1)}{1-\rho_1} \big( \rho_1^2 + \cot^2{\theta_1} \big)^{\frac12} \,,
\end{align}

with \(\rho_1 = \rho_1(\theta_1;v_2) \in (0,1)\) given by (<ref>).
Moreover, the pseudo-Mach number \(M_2^{(\theta_1)}\) is a strictly decreasing function of the incident angle \(\theta_1\)
and satisfies
that \(M_2^{(\theta_1)} \to \infty\) as \(\theta_1 \to 0^+\).

Formula (<ref>) follows directly from (<ref>) and (<ref>)–(<ref>).

For the second property,
the limit is clear since (<ref>) implies that \(\ell(\rho_1) \to -v_2 > 0\) as \(\theta_1 \to 0^+\),
whilst \(\cot^2{\theta_1} \to \infty\) as \(\theta_1 \to 0^+\).
To determine the strict monotonicity of \(M_2^{(\theta_1)}\), we differentiate (<ref>) and (<ref>)
with respect to \(\theta_1\) and re-arrange it to obtain

\begin{align}
\label{eq:appendixDerivativeExpressionMachNumberState2}
\frac{ (1-\rho_1)^2 \cot{\theta_1} }{2 ( \rho_1 + \cot^2{\theta_1} ) (\ell(\rho_1))^2} \frac{{\rm d} (M_2^{(\theta_1)})^2}{{\rm d} \theta_1} =
\rho_1 + \frac{1}{1-\rho_1}\frac{\ell(\rho_1)}{\ell'(\rho_1)} - \cot^2{\theta_1}\,.
\end{align}

Furthermore, from the definition of \(\ell(\rho)\) in (<ref>), we directly compute

\begin{align*}
(1 - \rho) \frac{\ell'(\rho)}{\ell(\rho)} = - \frac{(\ell(\rho))^2 + (1-\rho)^2 h'(\rho)}{(\ell(\rho))^2(1 + \rho)}\,.
\end{align*}

Substituting the above expression into the right-hand side
of (<ref>) gives

\begin{align*}
\text{RHS of~\eqref{eq:appendixDerivativeExpressionMachNumberState2}} =
\frac{ \rho_1 (1-\rho_1)^2 h'(\rho_1) - (\ell(\rho_1))^2}{(\ell(\rho_1))^2 + (1-\rho_1)^2 h'(\rho_1) } - \cot^2{\theta_1}\,.
\end{align*}

Denote the numerator of the first term above by

\begin{align*}
f(\rho) \defeq \rho (1-\rho)^2 h'(\rho) - (\ell(\rho))^2 \equiv \frac{1-\rho}{1+\rho} \, \tilde{f}(\rho)\,,
\end{align*}

where $\tilde{f}(\rho) \defeq (\gamma+1) h(\rho) - (\gamma-1) \rho^2 h(\rho) + (1 - \rho^2)$.
Observe that $\tilde{f}(1) = 0$ and

\begin{align*}
\tilde{f}'(\rho) = (\gamma + 1) (1-\rho^2) h'(\rho) > 0\, \qquad \mbox{for any $\rho\in(0,1)$}\,.
\end{align*}

Thus, \(\tilde{f}(\rho) < 0\) for all \(\rho\in(0,1)\).
It directly follows that the RHS of (<ref>) is negative.
We conclude that \(M_2^{(\theta_1)}\) is a strictly decreasing function of \(\theta_1 \in (0,\theta^{\rm cr})\).

It follows immediately from Lemma <ref> that \(\xi^{P_0^1} > 0\) is
a strictly decreasing function of \(\theta_1 \in (0,\theta^{\rm cr})\)
with \(\xi^{P_0^1} \to \infty\) as \(\theta_1 \to 0^+\),
\(M_2^{(\theta_1)} = \abs{( -\xi^{P_0^1}, v_2)}\).
Similarly, \(\xi^{P_0^2} < 0\) is a strictly increasing function of \(\theta_2 \in (0,\theta^{\rm cr})\)
with \(\xi^{P_0^2} \to -\infty\) as \(\theta_2 \to 0^+\).

We define the background solution $\bar{\varphi} \in C^{0,1}_{\rm loc}(\mathbb{R}^2_+)$ as

\begin{align}
\label{BackGroundSolu}
\bar{\varphi}(\bm{\xi}) \defeq \min\big\{ \varphi_1(\bm{\xi}), \varphi_2(\bm{\xi}), \varphi_3(\bm{\xi})\big\} \,,
\end{align}

and define the three open domains

\begin{equation}
\label{eq:domains-omega-123}
\Omega_i \defeq {\rm int}\,\{ \bm{\xi} \in \mathbb{R}^2_+ \,:\, \varphi_i(\bm{\xi}) = \bar{\varphi}(\bm{\xi})\}\,
\qquad \mbox{for $i = 1,2,3$}\,,
\end{equation}

where \({\rm int}\,G\) denotes the interior of a set \(G \subseteq \mathbb{R}^2\).
Note that \(\Omega_1\) and \(\Omega_3\) are the wedge-shaped domains with wedge
angles \(\theta_1\) and \(\theta_2\) respectively, and they become empty sets
in the limits: \(\theta_1 \to 0^+\) and \(\theta_2 \to 0^+\), respectively.

In light of the above discussion, we seek solutions of the following asymptotic boundary value problem
in the self-similar coordinates $\bm{\xi}=(\xi,\eta)$ in the upper half-plane $\mathbb{R}^2_+$.

[Asymptotic boundary value problem]
Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For any $\btheta \in (0,\theta^{\rm cr})^2,$ determine the existence of a weak
solution $\varphi\in W^{1,\infty}_{\rm loc}(\mathbb{R}_+^2)$ of
equations (<ref>)–(<ref>) in $\mathbb{R}_+^2$
satisfying the following conditions:

The asymptotic boundary condition at infinity:

\begin{equation*}
\lim_{r\to\infty} \norm{\varphi-\bar{\varphi}}_{C^{0,1}(R_{\theta} \setminus B_r(\bm{0}))} = 0\,
\qquad\, \text{for any $\theta \in (0,\pi)$}\,;
\end{equation*}

The slip boundary condition on the symmetric boundary:

\begin{equation*}
D\varphi\cdot(0,1)=0 \qquad \text{on $L_{\rm sym} = \big\{\bm{\xi} \in \mathbb{R}^2 \,:\, \eta=0\big\}$}\,.
\end{equation*}

§.§.§ Normal reflection configurations

It is meaningful to extend the range of parameters in Problem <ref> to allow \(\btheta \in [0,\theta^{\rm cr})^2\).
We study the reflection for the case: \(\btheta = \bm{0}\), which we call the normal reflection configuration.
It is clear that setting \(\varphi= \bar{\varphi} \equiv \varphi_2\) would satisfy Problem <ref>(<ref>)
at infinity.
However, since \(\varphi_2\) does not satisfy Problem <ref>(<ref>),
a boundary layer must be present.
For this reason, we introduce a uniform downstream state \((0)\), determined by a pseudo-potential \(\varphi_0\)
satisfying Problem <ref>(<ref>),
such that
a straight reflected shock \(S_0\) is formed between states \((2)\) and \((0)\).
In fact, the only possible straight reflected shock that satisfies <ref>(<ref>)
is a normal shock \(S_0 \defeq \{\bm{\xi} \in \mathbb{R}^2_+ \,:\, \eta = \eta_0 \}\) for some \(\eta_0 > 0\), which is parallel to \(L_{\rm sym}\).
The constant density of state \((0)\) given by \(\rho_0 \defeq \rho(\abs{D\varphi_0},\varphi_0)\) should satisfy the entropy condition \(\rho_0 > 1\).
We demonstrate that state \((0)\) described above is uniquely determined by \((\gamma, v_2)\).

The pseudo-potential \(\varphi_0\) has form (<ref>) with $i=0$,
for suitable constants \((u_0,v_0,k_0)\) to be determined.
It follows from Problem <ref>(<ref>) that
\(v_0 = 0\), whilst \(u_0 = 0\) follows from the Rankine–Hugoniot conditions (<ref>)
between states \((2)\) and \((0)\) because \(S_0\) is parallel to \(L_{\rm sym}\).
The value of \(k_0 = v_2 \eta_0\) is then obtained by using the continuity
of \(\varphi = \varphi_0 = \varphi_2\) on \(S_0\).

It remains to determine constant \(\eta_0 > 0\), which fixes the location of \(S_{0}\).
Combining the Bernoulli law (<ref>) with the Rankine–Hugoniot conditions (<ref>), we find the necessary condition

\begin{align} \label{eq:normal-reflection-density-relation}
&\ell(\rho_0) = -v_2\,.
\end{align}

It is clear that (<ref>) admits a unique solution \(\rho_0 \in (1 ,\infty)\)
depending only on \((\gamma, v_2)\).
Indeed, \(\ell(\cdot)\) is strictly increasing on \((1,\infty)\) and \(\ell(1) = 0\).
Constant \(\eta_0\) is then uniquely determined by the Rankine–Hugoniot conditions (<ref>) to be

\begin{equation}\label{Sol-NormalReflec0-RH}
{\eta}_0 \defeq -\frac{v_2}{\rho_0 - 1 } > 0\, \qquad \text{where \(\rho_0 \in (1,\infty)\)
\end{equation}

Thus, the pseudo-potential of state \((0)\) has been uniquely determined above by \((\gamma,v_2)\) as

\begin{align*}
\varphi_0 (\bm{\xi}) \defeq -\frac12 \abs{\bm{\xi}}^2 + v_2 \eta_0 \,.
\end{align*}

It is direct to verify that

\begin{align}
\label{eq:normal-reflection-solution}
\varphi_{\rm norm}(\bm{\xi}) \defeq \min\{\varphi_0(\bm{\xi}), \varphi_2(\bm{\xi})\} =
\begin{cases}
\varphi_2(\bm{\xi}) &\quad \text{for}\; \eta > \eta_0\,, \\
\varphi_0(\bm{\xi}) \, \quad &\quad \text{for}\; 0 < \eta < \eta_0\,,
\end{cases}
\end{align}

is a solution to Problem <ref> when \(\btheta = \bm{0}\); see Fig. <ref>(a).

Briefly, we mention the case:
\(\btheta \in \big( \{0\} \times (0,\theta^{\rm cr}) \big) \cup \big( (0,\theta^{\rm cr}) \times \{0\}\big)\),
which is called the unilateral normal reflection; a normal reflection on one side with respect to
the symmetric boundary $L_{\rm sym}$ occurs by the same argument as the normal reflection
case $\btheta=\mathbf{0}$ above, while the other side undergoes a regular reflection at the reflection point,
of which details will be discussed in <ref> below;
see the case: $\btheta \in (0,\theta^{\rm cr})\times \{0\}$ in Fig. <ref>(b).
Moreover, the unilateral normal reflections are essentially the same as the Prandtl-Meyer reflection
configurations considered in [2].

*(a) Normal reflection when $\theta_1=\theta_2={0}$.

*(b) Unilateral normal reflection when $\theta_1>\theta_2=0$.

Structure of solutions of Problem <ref> involving normal shocks.

§.§ Regular reflection configurations

In this section, we introduce the detachment and sonic angles and
describe the structure of three genuinely different configurations of the four-shock interactions.

§.§.§ Detachment and sonic angles

For \(\btheta \in (0,\theta^{\rm cr})^2\), the incident shocks \(S_{12}\) and \(S_{32}\) intersect
the symmetric line \(L_{\rm sym}\) at points \(P_0^1\) and \(P_0^2\) respectively,
with coordinates given by (<ref>).
As discussed above, the symmetric line can be regarded as a rigid wall and, similarly to
the previous works [10], we expect that, if the incident angle $\theta_{1}$ is
less than a detachment angle $\theta^{\rm d}$,
there exists a regular reflection of \(S_{12}\) at point \(P_0^1\), i.e.,
there exist both a uniform state \((5)\) determined by a pseudo-potential \(\varphi_5\)
and a straight reflected shock \(S_{25}\) passing through \(P_0^1\)
such that \(S_{25}\) separates the uniform upstream state \((2)\) from the uniform downstream state \((5)\).
Similarly, we expect that there exists a regular reflection of \(S_{32}\) at point \(P_0^2\)
if $\theta_2$ is less than $\theta^{\rm d}$;
that is, there exist both a uniform state \((6)\) determined by a pseudo-potential \(\varphi_6\)
and a straight reflected shock \(S_{26}\) passing through \(P_0^2\).
The expected structure of shock regular reflection on the upper half-plane
can be seen in Fig. <ref>.
We now determine the detachment angle $\theta^{\rm d}$ and the sonic angle $\theta^{\rm s}$.

The regular reflection of \(S_{12}\) at point \(P_0^1\) can be reduced to the study of an algebraic system.
By the argument in <cit.>,
to determine the uniform state \((5)\), it suffices to apply the Rankine–Hugoniot conditions
at the reflection point \(P_0^1\) only.
The pseudo-potential \(\varphi_5\) has form (<ref>) with $i=5$, and the straight reflected shock \(S_{25}\)
must have the form:

\begin{align*}
S_{25} \defeq \big\{ \bm{\xi} \in \cl{\mathbb{R}^2_+} \,:\, \varphi_5(\bm{\xi})
= \varphi_2(\bm{\xi})\big\} = \big\{ \bm{\xi} \in \cl{\mathbb{R}^2_+} \,:\,
\eta = \xi \tan{\theta_{25}} + a_{25}
\big\}
\end{align*}

for some constants \((u_5,v_5,k_5, \theta_{25}, a_{25})\) depending on \((\gamma, v_2, \theta_1)\) to be determined,
where \(\theta_{25}\) is the angle between the reflected shock \(S_{25}\) and the positive \(\xi\)–axis, and \(a_{25}\)
is the \(\eta\)–intercept of the reflected shock \(S_{25}\).
Applying the Rankine–Hugoniot conditions (<ref>) between states \((2)\) and \((5)\) at \(P_0^1\), we have

\begin{align} \label{eq:state5RHcondition1}
\begin{split}
&\varphi_5(P_0^1) = \varphi_2(P_0^1)\,, \;\;
D\varphi_5(P_0^1) \cdot \bm{\tau}_{25} = D \varphi_2(P_0^1) \cdot \bm{\tau}_{25}\,,
\;\; \rho_5 D\varphi_5(P_0^1) \cdot \bm{\nu}_{25}  = D \varphi_2(P_0^1) \cdot \bm{\nu}_{25}\,,
\end{split}
\end{align}

where \(\bm{\nu}_{25} \defeq \frac{D(\varphi_2 - \varphi_5)(P_0^1)}{|D(\varphi_2 - \varphi_5)(P_0^1)|}\), \(\bm{\tau}_{25}
\defeq \bm{\nu}^\perp\), and \(\rho_5 \defeq \rho( |D\varphi_5|,\varphi_5 )\) is given by (<ref>).
Furthermore, the pseudo-potential \(\varphi_5\) should satisfy the entropy condition:

\begin{align} \label{eq:state5entropycondition1}
\rho_5 > 1\,,
\quad \text{or equivalently}\,, \quad
D \varphi_2 (P_0^1) \cdot \bm{\nu}_{25} > D \varphi_5(P_0^1) \cdot \bm{\nu}_{25} > 0\,;
\end{align}

and the slip boundary condition:

\begin{equation} \label{eq:state5slipBC}
D\varphi_5 \cdot (0,1) = 0 \qquad \text{on $L_{\rm sym}$}\,.
\end{equation}

A necessary condition for the regular reflection of \(S_{12}\) at \(P_0^1\) is clearly
that state \((2)\) must be pseudo-supersonic at \(P_0^1\).
From Lemma <ref>,
there exists a unique \(\theta^+ \in (0,\theta^{\rm cr}]\) such that \(M_2^{(\theta_1)} > 1\) if and only if \(\theta_1 \in (0,\theta^+)\).
However, note that \(M_2^{(\theta_1)}|_{\theta_1 = \theta^+}\) is not necessarily $1$ if \(\theta^+=\theta^{\rm cr}\).

Introduce the steady detachment angle \(\theta_{\rm stdy}^{\rm d}(\rho_\infty,u_\infty)\)
for the steady potential flow with constant supersonic upstream state \(U_\infty = (\rho_\infty,u_\infty,0)\),
as defined in <cit.>.
Also, for \(\theta_1 \in (0,\theta^{\rm cr})\),
denote by \(\hat{\theta}_{25}(\theta_1)\) the acute angle between the pseudo-velocity \(D\varphi_2(P_0^1) = (-\xi^{P_0^1}, v_2)\)
and the symmetric line \(L_{\rm sym}\):

\begin{align} \label{eq:state5SelfSimilarTurningAngle}
\hat{\theta}_{25}(\theta_1) \defeq \arccos{(
\frac{\xi^{P_0^1}}
= \arcsin{( \frac{- v_2}{|D\varphi_2(P_0^1)|}) } \in ( 0, \tfrac{\pi}{2} )\,.
\end{align}

Similarly to <cit.>, whenever \(\theta_1 \in (0 ,\theta^+)\),
the existence and multiplicity of solutions of the algebraic system (<ref>)–(<ref>)
are determined as follows:

* If \(\hat{\theta}_{25}(\theta_1) < \theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1)|)\), there are two solutions;

* If \(\hat{\theta}_{25}(\theta_1) = \theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1)|)\), there is one solution;

* If \(\hat{\theta}_{25}(\theta_1) > \theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1)|)\), there are no solutions.

We demonstrate that there exists at most one value \(\theta^{\rm d} \in (0,\theta^+)\)
such that \(\theta_1 = \theta^{\rm d}\) satisfies the equality in (b) above.
Indeed, using (<ref>) and Lemma <ref>,
it is clear that \(\hat{\theta}_{25}(\theta_1)\) is a strictly increasing function
of \(\theta_1\in (0 ,\theta^{\rm cr})\), whilst,
for \(\theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1)|)\), we state below a general fact
about 2-D steady potential flow, from which it is clear that \(\theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1)|)\)
is strictly decreasing with \(\theta_1 \in (0,\theta^+)\).
The proof of this statement is given in Appendix <ref>.

Let \(\theta_{\rm stdy}^{\rm d}\) and \(\theta_{\rm stdy}^{\rm s}\) denote the steady detachment and sonic angles
for the steady potential flow with constant supersonic upstream state \(U_\infty = (\rho_\infty,u_\infty,0)\).
Then \(\theta_{\rm stdy}^{\rm d}\) and \(\theta_{\rm stdy}^{\rm s}\) are smooth, strictly
increasing functions of the upstream Mach number \(M_\infty \defeq \frac{u_\infty}{c(\rho_\infty)} > 1\).
Moreover, the following limits hold:

\begin{align*}
\lim_{M_\infty \to 1^+} (\theta^{\rm d}_{\rm stdy}, \theta^{\rm s}_{\rm stdy}) = (0 , 0) \,, \qquad\,\,
\lim_{M_\infty \to \infty} (\theta^{\rm d}_{\rm stdy}, \theta^{\rm s}_{\rm stdy})
= (\tfrac{\pi}{2}, \arctan{\sqrt{\tfrac{2}{\gamma-1}}})\,.
\end{align*}

With the above discussion, the following proposition gives the necessary and sufficient criterion
on the incident angle $\theta_1$ for the existence of a regular reflection of \(S_{12}\) at point \(P_0^1\).

Fix \(\gamma\geq 1\) and \(v_2 \in (v_{\min},0)\).
There exists a unique \(\theta^{\rm d} = \theta^{\rm d}(\gamma, v_2) \in (0, \theta^{\rm cr} ],\)
called the detachment angle, such that,
for \(\theta_1 \in (\theta^{\rm d},\theta^{\rm cr}),\) there are no states \((5)\) of form (<ref>)
satisfying (<ref>)–(<ref>) and,
for each \(\theta_1 \in (0,\theta^{\rm d}),\) there are exactly two states \((5)\) of form (<ref>)
satisfying (<ref>)–(<ref>):
the weak reflection state \(\varphi_5^{\rm wk}\) and the strong reflection state \(\varphi_5^{\rm sg},\)
distinguished by
\(1 < \rho_5^{\rm wk} < \rho_5^{\rm sg}\,,\)
where \(\rho_5^{\rm wk} \defeq \rho(\abs{D\varphi_5^{\rm wk}},\varphi_5^{\rm wk})\)
and \(\rho_5^{\rm sg} \defeq \rho(\abs{D\varphi_5^{\rm sg}},\varphi_5^{\rm sg})\)
are the constant densities of the weak and strong reflection states \((5)\) respectively.
Furthermore, denoting by
\(O_5^{\rm wk} \defeq (u_5^{\rm wk}, v_5^{\rm wk})\) and \(O_5^{\rm sg} \defeq (u_5^{\rm sg}, v_5^{\rm sg})\) the sonic centres,
and by \(c_5^{\rm wk}\) and \(c_5^{\rm sg}\) the sonic speeds of the weak and strong reflection states \((5)\) respectively,

\((\rho_5^{\rm wk}, O_5^{\rm wk})\in C([0,\theta^{\rm d}])\cap C^{\infty}([0,\theta^{\rm d}))\) and
\((\rho_5^{\rm sg}, O_5^{\rm sg}) \in C((0 ,\theta^{\rm d}])\cap C^{\infty}((0,\theta^{\rm d}))\).
In particular, the following limits exist:

\begin{align*}
\lim_{\theta_1 \to 0^+} (\rho_5^{\rm wk}, O_5^{\rm wk}, u_5^{\rm wk} \xi^{P_0^1}) = (\rho_0, \mathbf{0}, -v_2 \eta_0)\,,
\qquad
\lim_{\theta_1 \to \theta^{\rm d-}} (\rho_5^{\rm wk}, \rho_5^{\rm sg}, O_5^{\rm wk}, O_5^{\rm sg})\,,
\end{align*}

where \(\rho_0\) and \(\eta_0\) are the density and shock location of the normal reflection state $(0)$
which are given in <ref>.

For any \(\theta_1 \in (0,\theta^{\rm d}),\)

\begin{align}
&| D\varphi_2 (P_0^1) | > | D\varphi_2 (P_0^1|_{\theta_1 = \theta^{\rm d}}) | > 1\,,\label{eq:state2SupersonicCondition}\\
&0 < u_5^{\rm wk} < u_5^{\rm sg} < \xi^{P_0^1}\,, \qquad  v_5^{\rm wk } = v_5^{\rm sg} = 0\,. \label{eq:localtheorystate5-sonic-centres}
\end{align}

There exists a unique \(\theta^{\rm s} = \theta^{\rm s}(\gamma, v_2) \in (0, \theta^{\rm d}],\) called the sonic angle, such that

| D φ_5^wk(P_0^1) | > c_5^wk            for all    θ_1 ∈(0, θ^s) ,

| D φ_5^wk(P_0^1) | < c_5^wk            for all    θ_1 ∈(θ^s, θ^d] .

Furthermore, if \(\theta^{\rm s} < \theta^{\rm cr},\) then \(\theta^{\rm s} < \theta^{\rm d}\)
and $| D \varphi_5^{\rm wk}(P_0^1) | = c_5^{\rm wk}$ when $\theta_1 = \theta^{\rm s}$.

The proof
follows similarly to <cit.>, once $\theta^{\rm d}$ and $\theta^{\rm s}$ are determined.
Thus, it remains to show the existence and uniqueness of $\theta^{\rm d}$ and $\theta^{\rm s}$.

As discussed above, \(\hat{\theta}_{25}(\theta_1)\) is strictly increasing with \(\theta_1\in (0 ,\theta^{\rm cr})\),
whilst \(\theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1)|)\) is strictly decreasing with \(\theta_1 \in (0,\theta^+)\).
Furthermore, Lemmas <ref> and <ref> show that

\begin{align*}
\lim_{\theta_1\to 0^+} \hat{\theta}_{25}(\theta_1)
= 0 \,\,
< \,\, \tfrac\pi2
= \lim_{\theta_1\to 0^+} \theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1)|)\,.
\end{align*}

Then it follows that there exists at most one solution \(\vartheta^{\rm d} \in (0, \theta^+)\) of the equation:

\begin{align} \label{eq:uniquenessOfThetaD}
\hat{\theta}_{25}(\vartheta^{\rm d}) = \theta_{\rm stdy}^{\rm d}(1,|D\varphi_2(P_0^1|_{\theta_1 = \vartheta^{\rm d}})|)\,.
\end{align}

We define the detachment angle \(\theta^{\rm d} \in (0 , \theta^+]\) by

\begin{align*}
\theta^{\rm d} \defeq
\min\big\{\theta^+, \,\inf\{ \vartheta^{\rm d} \in (0,\theta^+) \,:\,
\text{\(\vartheta^{\rm d}\) satisfies~\eqref{eq:uniquenessOfThetaD}}\} \big\}\,.
\end{align*}

In this sense, \(\theta^{\rm d}\) may refer to either the conventional detachment angle \(\vartheta^{\rm d}\)
that satisfies equation (<ref>) when such a solution exists, or angle \(\theta^+\) otherwise.

The existence and properties of the sonic angle \(\theta^{\rm s}\) contained in statement (<ref>)
of Proposition <ref> follow from a similar argument as above.
In particular, symbol \(\theta^{\rm s}\) may refer to either the conventional
sonic angle \(\vartheta^{\rm s} \in (0,\theta^{\rm d}]\) which should satisfy

\begin{align} \label{eq:uniqueness-of-theta-s}
\hat{\theta}_{25}(\vartheta^{\rm s}) = \theta^{\rm s}_{\rm stdy}(1, \abs{D\varphi(P_0^1|_{\theta_1 = \vartheta^{\rm s}})})\,;
\end{align}

or we simply define \(\theta^{\rm s} = \theta^{\rm d}\) if no such solution exists.
That is, we define

θ^s min{θ^d,  inf{ϑ^s ∈(0,θ^d)  :   \(\vartheta^{\rm s}\) satisfies (<ref>) }} .

We have the following remarks about Proposition <ref>:

Fix \(\gamma \geq 1\).
For any $v_2\in(\vmin,0),$ let \(\theta^{\rm cr}\) be the critical angle given by (<ref>),
and let \(v_2^{\rm s},v_2^{\rm d}\in(\vmin,0)\) be the constants given by Lemma <ref>.

\begin{align*}
\theta^{\rm s} = \theta^{\rm d} = \theta^{\rm cr} \quad &\Longleftrightarrow \quad v_2 \in (v_{\min}, v_2^{\rm s}]\,,\\
\theta^{\rm s} < \theta^{\rm d} = \theta^{\rm cr} \quad &\Longleftrightarrow \quad v_2 \in (v_2^{\rm s}, v_2^{\rm d}]\,,\\
\theta^{\rm s} < \theta^{\rm d} < \theta^{\rm cr} \quad &\Longleftrightarrow \quad v_2 \in (v_2^{\rm d}, 0)\,.
\end{align*}

By the symmetry of the problem, the conclusions of Proposition <ref> immediately
apply to the regular reflection of \(S_{32}\) at \(P_0^2\) after exchanging \((\theta_1,P_0^1,(5))\) with \((\theta_2,P_0^2,(6)),\)
and reversing the inequalities in (<ref>).
In particular, the same detachment angle $\theta^{\rm{d}}(\gamma, v_2)$ and
sonic angle $\theta^{\rm{s}}(\gamma, v_2)$ apply to the incident angle $\theta_2$ and state \((6)\).

The main difference between Proposition <ref> and <cit.>
is that we assert the uniqueness of solutions to (<ref>) and (<ref>),
although we cannot in general
assert the existence of such solutions due to the restriction \(\theta_1 \in (0,\theta^{\rm cr})\);
whereas <cit.> asserts the existence,
but not the uniqueness of such solutions.

Fix \(\gamma \geq 1\) and \(v_2 \in (v_{\min},0)\).
Let ${\theta}^{\rm{s}}\defeq{\theta}^{\rm{s}}(\gamma,v_2)$ and ${\theta}^{\rm{d}} \defeq {\theta}^{\rm{d}}(\gamma,v_2)$
be given by Proposition <ref>.
Denote the set of admissible parameters

\begin{equation*}
\Theta \defeq [0,\theta^{\rm d})^2 \setminus \{\bm{0}\}
\subseteq [0,\theta^{\rm cr}]^2\,.
\end{equation*}

Hereafter, for \(\btheta \in \cl{\Theta},\)  states \((5)\) and \((6)\) refer to the weak reflection states \((5)\) and \((6)\)
from Proposition <ref>.
For \(j = 5, 6,\) we write \((\rho_j,c_j,u_j,v_j) \defeq (\rho_j^{\rm wk},c_j^{\rm wk},u_j^{\rm wk},v_j^{\rm wk}),\) \(O_j \defeq (u_j,v_j),\)
and \(\varphi_{j} \defeq \varphi_j^{\rm wk}\).
The pseudo-potential of state \((j)\) is given by

\begin{equation} \label{eq:def-weak-state-5-6}
\varphi_j (\bm{\xi}) = -\frac12 \abs{\bm{\xi}}^2 + (u_j,v_j) \cdot \bm{\xi} + v_2 a_{2j}\,,
\end{equation}

where $a_{2j} \defeq - \frac{1}{v_2} \xi^{P_0^{j-4}} u_j$ for $\theta_{j-4} > 0$,
and $a_{2j} \defeq \eta_0$ when $\theta_{j-4} = 0$ $($cf. Proposition 2.6(i)$)$.

Define the reflected shocks \(S_{25}\) and \(S_{26}\) by

\begin{equation*}
S_{2j} \defeq
\big\{\bm{\xi}\in\cl{\mathbb{R}^2_+} \,:\, \varphi_j(\bm{\xi}) = \varphi_2(\bm{\xi})\big\}
= \big\{ \bm{\xi} \in \cl{\mathbb{R}^2_+} \,:\,
\eta = \xi \tan \theta_{2j}   +   a_{2j}
\big\} \qquad\text{for}\;\; j=5,6\,,
\end{equation*}

where \(\theta_{25}\) and \(\theta_{26}\) are the angles between the reflected shocks \(S_{25}\) and \(S_{26}\)
and the positive \(\xi\)–axis, given respectively by

\begin{align} \label{eq:def-reflected-shock-angle-25-26}
\theta_{25} \defeq \arctan{\big(\frac{u_5}{v_2}\big)} \in (\tfrac{\pi}{2},\pi]\,,
\qquad \theta_{26} \defeq \arctan{\big( \frac{u_6}{v_2} \big)} \in [0 ,\tfrac{\pi}{2} )\,.
\end{align}

The unit tangent vectors to \(S_{25}\) and \(S_{26}\) are given respectively by

\begin{align} \label{eq:tangent-vectors-e25-e26}
\bm{e}_{S_{25}}  \defeq (\cos\theta_{25}, \sin\theta_{25})\,, \qquad
\bm{e}_{S_{26}} \defeq (\cos\theta_{26}, \sin\theta_{26})\,.
\end{align}

It follows from
Proposition <ref>(<ref>)
and Definition <ref> that
\(\varphi_j = \varphi_0\), and $\theta_{2j}=(6-j)\pi$ when \(\theta_{j-4} = 0\) for \(j = 5,6\).

Fix \(\gamma \geq 1\) and \(v_2 \in (v_{\min}, 0 )\).
Let \(\rho_5\) and \(\rho_6\) be the constant densities of states \((5)\) and \((6)\) given
by Proposition <ref>, and let \(\theta_{25}\) and \(\theta_{26}\)
be the reflected shock angles given by (<ref>).
Then, for \(j = 5,6,\) there exists a constant \(\delta_{5,6} > 0\) depending only on \((\gamma, v_2)\)
such that, for any \(\btheta \in \cl{\Theta},\)

\begin{align}
&1 + \delta_{5,6} \leq \rho_j \leq \delta_{5,6}^{-1}\,,\label{eq:density-state-5-6-uniform-bounds} \\
&\delta_{5,6} \leq \abs{\theta_{2j} - \tfrac\pi2}\,.\label{eq:reflected-shock-angle-upper-bound}
\end{align}

Moreover,  for any \(\bar{\theta} \in (0,\theta^{\rm d})\), there exists
\(\delta_{5,6}^{(\bar{\theta})} > 0\)
depending only on \((\gamma, v_2, \bar{\theta})\) such that

\begin{align}
\label{eq:reflected-shock-angle-lower-bound}
\delta_{5,6}^{(\bar{\theta})} &< \abs{\theta_{2j} - (6-j)\pi}
\qquad \mbox{whenever $\btheta \in \cl{\Theta} \cap \{\theta_{j-4} > \bar{\theta}\}$}\,.
\end{align}

We prove the results only for state \((5)\), since the proof for state \((6)\) is similar.

Fix \(\gamma \geq 1\) and \(v_2 \in (v_{\min},0)\).
Proposition <ref>(<ref>),
we see that \(\rho_5\in C(\cl{\Theta})\), so that the uniform upper bound
in (<ref>) is clear.
For the uniform lower bound, the entropy condition (<ref>) gives \(\rho_5 > 1\).
Furthermore, the Rankine–Hugoniot conditions between states \((2)\) and \((5)\) imply that

\begin{equation}\label{Eq:LengthO2O5CompatiCondi}
\ell(\rho_5) = \big|(u_2,v_2) - (u_5,v_5)\big|
= \sqrt{v_2^2 + u_5^2} \geq |v_2| > 0\, \qquad \text{for any $\btheta \in \cl{\Theta}$}\,,
\end{equation}

with  \(\ell(\cdot)\) given by (<ref>).
In particular, \(\ell(\cdot)\) is an increasing function on \([1,\infty)\) with \(\ell(1) = 0\),
and hence (<ref>) implies the existence of a small constant
\(\delta^{(1)}_{5,6} > 0\), depending only on \((\gamma, v_2)\),
such that \(\rho_5 > 1 + \delta^{(1)}_{5,6}\) for any \(\btheta \in \cl{\Theta}\).

Next, we prove (<ref>).
From (<ref>) and the geometric properties of the shock polar curve for 2-D
steady potential flow, we have

\begin{align} \label{eq:simple-reflection-angle-estimate}
\tfrac{\pi}{2} + \hat{\theta}_{25} < \theta_{25} \leq \pi \,
\qquad\, \text{for any $\btheta \in \cl{\Theta}$}\,,
\end{align}

with \(\hat{\theta}_{25}\) given by (<ref>).
From Proposition <ref>(i) and (<ref>),
we see that \(\theta_{25} \to \pi^- \) as \(\theta_1 \to 0^+\).
By continuity, there exists a small constant \(\delta_{5,6}^{(2)} > 0\) depending only on \((\gamma, v_2)\)
such that \(\theta_{25} > \frac{3\pi}{4}\) whenever \(\theta_1 \in [ 0 , \delta_{5,6}^{(2)} ) \).
On the other hand, by (<ref>) and Lemma <ref>,
there exists a constant \(\delta_{5,6}^{(3)} > 0 \) depending only on \((\gamma, v_2)\)
such that \(\hat{\theta}_{25} > \delta_{5,6}^{(3)}\) whenever \( \theta_1 \in [\delta_{5,6}^{(2)}, \theta^{\rm d}]\).
Together with (<ref>), we conclude
that \(\theta_{25} > \min\{\frac{3\pi}{4}, \frac{\pi}{2} + \delta_{5,6}^{(3)}\}\) for all \(\theta_1 \in [0, \theta^{\rm d}]\),
which leads to (<ref>).

Finally, bound (<ref>) follows directly
from (<ref>) and Lemma <ref>,
which states that \(\theta_{25}\) is a strictly decreasing function of \(\theta_1 \in [0, \theta^{\rm d}]\).

§.§.§ Configurations for four-shock interactions

The four-shock interaction configurations are split into three genuinely different cases,
depending on parameters \(\btheta \in \Theta\) and the critical angles \(\theta^{\rm s}\) and \(\theta^{\rm d}\).
We briefly describe the expected structure of each configuration and introduce some important notation.

I Supersonic-supersonic reflection configuration.
When \(\btheta \in (0,{\theta}^{\rm s})^2\), the supersonic-supersonic reflection configuration is expected; see Fig. <ref>.
In this case, both states \((5)\) and \((6)\) are pseudo-supersonic at the reflection points $P_0^1$ and $P_0^2$,

I The supersonic-supersonic reflection configuration.

II Supersonic-subsonic reflection configuration.
When \(\btheta \in [\theta^{\rm{s}},\theta^{\rm{d}}) \times (0,\theta^{\rm{s}})\),
or \(\btheta \in (0,\theta^{\rm{s}}) \times [\theta^{\rm{s}},\theta^{\rm{d}})\), the supersonic-subsonic reflection configuration is expected;
see Fig. <ref> for the case: \(\btheta \in [\theta^{\rm{s}},\theta^{\rm{d}}) \times (0,\theta^{\rm{s}})\).
In this case, either state \((5)\) is pseudo-sonic/pseudo-subsonic at $P_0^1$ and state \((6)\) is pseudo-supersonic at $P_0^2$, or vice versa.

II The supersonic-subsonic reflection configuration in the case:
\(\btheta \in [\theta^{\rm{s}},\theta^{\rm{d}}) \times (0,\theta^{\rm{s}})\), i.e., state \((5)\) is pseudo-sonic/pseudo-subsonic at \(P_0^1\).

III Subsonic-subsonic reflection configuration.
When \(\btheta \in [\theta^{\rm s},\theta^{\rm d})^2\), the subsonic-subsonic reflection configuration is expected;
see Fig. <ref>. In this case, both states $(5)$ and $(6)$ are pseudo-subsonic at the reflection points $P_0^1$ and $P_0^2$, respectively.

It follows from Remark <ref>(<ref>) that the configurations of Case II and Case III
are not possible unless \(v_2 \in (v_2^{\rm s},0)\), for constant \(v_2^{\rm s}\) from Lemma <ref>.

III The subsonic-subsonic reflection configuration.

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For any $\btheta \in\cl{\Theta}$, let $Q_5$ and $Q_6$ be the orthogonal projections of \(O_5\) and \(O_6\) onto \(S_{25}\) and \(S_{26},\) respectively.
When \(\theta_1 \in (0,\theta^{\rm s}),\) the right sonic boundary and intersection points are given by

\begin{equation*}
\begin{aligned}
&\Gamma_{\rm sonic}^5\defeq
\big\{\bm{\xi}\in\partial B_{c_5}(O_5) \cap \overline{\mathbb{R}^2_+} \,:\, \xi^{Q_5} \leq \xi \leq \xi^{P_0^1},\, \varphi_5(\bm{\xi}) \leq \varphi_2(\bm{\xi})\big\}\,,\\
&\{P_1\} \defeq \Gamma_{\rm sonic}^5 \cap L_{\rm sym}\,, \quad
\{P_2\} \defeq \Gamma_{\rm sonic}^5 \cap S_{25}\,,
\end{aligned}
\end{equation*}

whilst, for \(\theta_1 \in [\theta^{\rm s},\theta^{\rm d}),\)

\begin{align*}
\Gamma_{\rm sonic}^5 \equiv \{P_1\} \equiv \{P_2\} \defeq \{P_0^1\}\,.
\end{align*}

Similarly, when \(\theta_2 \in (0,\theta^{\rm s}),\) the left sonic boundary and intersection points are given by

\begin{equation*}
\begin{aligned}
&\Gamma_{\rm sonic}^6\defeq \big\{\bm{\xi}\in\partial B_{c_6}(O_6) \cap \overline{\mathbb{R}^2_+} \,:\, \xi^{P_0^2}
\leq \xi \leq \xi^{Q_6}, \, \varphi_6(\bm{\xi}) \leq \varphi_2(\bm{\xi}) \big\}\,,\\
&\{P_3\} \defeq \Gamma_{\rm sonic}^6 \cap S_{26}\,,
\quad \{P_4\} \defeq \Gamma_{\rm sonic}^6 \cap L_{\rm sym}\,,
\end{aligned}
\end{equation*}

whilst, for \(\theta_2 \in [\theta^{\rm s},\theta^{\rm d}),\)

\begin{align*}
\Gamma_{\rm sonic}^6 \equiv \{P_3\} \equiv \{P_4\} \defeq \{P_0^2\} \,.
\end{align*}

The open regions are given by

\begin{equation*}
\begin{aligned}
\Omega_5 &\defeq \big\{ \bm{\xi} \in \mathbb{R}^2_+ \setminus \cl{B_{c_5}(O_5)} \,:\, \varphi_5(\bm{\xi}) < \varphi_2(\bm{\xi}),
\, \xi > \xi^{P_2} \big\}\,,\\
\Omega_6 &\defeq \big\{ \bm{\xi} \in \mathbb{R}^2_+ \setminus \cl{B_{c_6}(O_6)} \,:\, \varphi_6(\bm{\xi}) < \varphi_2(\bm{\xi}), \,
\xi < \xi^{P_3} \big\}\,.
\end{aligned}
\end{equation*}

Finally, define the symmetric boundary

\begin{align*}
\Gamma_{\rm sym} \defeq \big\{ (\xi, 0)\in L_{\rm sym} \,:\, \xi^{P_4} < \xi < \xi^{P_1}\big\}\,,
\end{align*}

and the line segments

\begin{equation*}
S^{\rm seg}_{25} \defeq S_{25} \cap \big\{ \xi^{P_2} \leq \xi \leq \xi^{P_0^1}\big\}\,,
\qquad
S^{\rm seg}_{26} \defeq S_{26} \cap \big\{ \xi^{P_0^2} \leq \xi \leq \xi^{P_3}\big\}\,.
\end{equation*}

Note that curves $\Gamma_{\rm{sonic}}^5$, $\Gamma_{\rm{sonic}}^6$, and $\cl{\Gamma_{\rm{sym}}}$
as defined in Definition <ref> above are fixed by \((\gamma,v_2,\btheta)\)
and do not share any common points except their endpoints $\{P_1,P_4\}$.
Also, note that $\Omega_j=\varnothing$ if $\theta_{j-4}\in[\theta^{\rm s},\theta^{\rm d})$ for $j=5,6$.
For a suitable simple open curve segment \(\Gamma_{\rm shock}\) (yet to be determined) with
endpoints \(P_2\) and \(P_3\),
we use $\Omega$ to denote the open bounded domain enclosed by the simple closed curve
$\Gamma_{\rm shock}\cup\Gamma_{\rm sonic}^5\cup\Gamma_{\rm sym}\cup\Gamma_{\rm sonic}^6$;
see Figs. <ref>–<ref>.

§.§ Admissible solutions and main theorems

We reformulate the asymptotic boundary value problem, Problem <ref>,
into a free boundary problem.
Our goal is to find a suitable curved free boundary \(\Gamma_{\rm shock}\) between points \(P_2\) and \(P_3\),
and a solution \(\varphi\) of Problem <ref> which satisfies (<ref>)–(<ref>)
in the open region \(\Omega\) (as depicted in Figs. <ref>–<ref>)
and the corresponding boundary conditions
on $\partial\Omega\defeq\Gamma_{\rm shock}\cup\Gamma^5_{\rm sonic}\cup\Gamma_{\rm sym}\cup\Gamma^6_{\rm sonic}$.
Furthermore, equation (<ref>) should be elliptic in \(\Omega\),
and the free boundary \(\Gamma_{\rm shock}\) should be a transonic shock that separates the hyperbolic
and elliptic regions of the solution.

[Free boundary problem]
Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For any $\btheta \in\overline{\Theta},$ find a free boundary $($curved reflected-diffracted shock$)$
$\Gamma_{\rm{shock}}$ in region $\mathbb{R}^2_+\cap\{\xi^{P_3}<\xi<\xi^{P_2}\}$ and a function $\varphi$
defined in region $\Omega,$ as shown
in Figs. <ref>–<ref>, such that $\varphi$ satisfies:

*  Equation (<ref>) in \(\Omega\)
so that (<ref>) is elliptic inside \(\Omega\);

*  $\varphi=\varphi_2$ and $\rho D\varphi\cdot\bm{\nu}=\rho_2 D\varphi_2\cdot\bm{\nu}$
on the free boundary $\Gamma_{\rm{shock}}$;

*  $\varphi=\varphi_j$ and $D\varphi=D\varphi_j$ on $\Gamma_{\rm{sonic}}^j$ for $j=5,6$;

$D\varphi\cdot(0,1)=0$ on $\Gamma_{\rm{sym}},$

where $\bm{\nu}$ is the unit normal vector on $\Gamma_{\rm{shock}}$ pointing into $\Omega$.

Fix $\gamma\geq1$ and $v_2\in(\vmin,0)$. For any $\btheta \in\overline{\Theta},$ a function $\varphi\in W^{1,\infty}_{\rm loc}(\mathbb{R}^2_+)$
is called an admissible solution to the four-shock interaction problem if $\varphi$
is a solution of Problem <ref> and satisfies the following properties:

*  There exists a relatively open curve segment $\Gamma_{\rm shock}$ $($without self-intersection$)$
with endpoints $P_2=(\xi^{P_2},\eta^{P_2})$ and $P_3=(\xi^{P_3},\eta^{P_3})$
given by Definition <ref>
such that $\Gamma_{\rm shock}$ is $C^2$ in its relative interior,
$\Gamma_{\rm shock}^{\rm ext}\defeq S^{\rm seg}_{25} \cup \Gamma_{\rm{shock}} \cup S^{\rm seg}_{26}$
is a \(C^1\)–curve including at \(P_2\) and \(P_3,\) with $S^{\rm seg}_{25}$ and $S^{\rm seg}_{26}$
given in Definition <ref>,

\begin{equation*}
\Gamma_{\rm shock} \subseteq
\big\{\bm{\xi}\in \mathbb{R}^2_+ \setminus \cl{B_{c_2}(O_2)} \,:\, \xi^{P_3} < \xi < \xi^{P_2} \big\} \,,
\end{equation*}

with $\partial B_{c_2}(O_2)$ as the sonic circle of state \((2)\),
curves $\cl{\Gamma_{\rm shock}}$ and $\Gamma_{\rm sonic}^5 \cup \Gamma_{\rm sym} \cup \Gamma_{\rm sonic}^6$
share no common points except their endpoints $\{P_2,P_3\}$ so that
$\Gamma_{\rm shock} \cup \Gamma_{\rm sonic}^5 \cup \Gamma_{\rm sym} \cup \Gamma_{\rm sonic}^6$ is a closed curve without self-intersection.

For the open bounded domain $\Omega\subseteq\mathbb{R}_+^2$
by $\Gamma_{\rm shock} \cup \Gamma_{\rm sonic}^5 \cup \Gamma_{\rm sym} \cup \Gamma_{\rm sonic}^6,$
solution $\varphi$ is in
$C^3(\Omega)\cap C^2\big(\overline{\Omega}\setminus(\Gamma_{\rm{sonic}}^5 \cup \Gamma_{\rm{sonic}}^6)\big)\cap C^1(\overline{\Omega})$
and satisfies $\varphi \in C^{0,1}_{\rm loc}(\mathbb{R}^2_+)
\cap C_{\rm loc}^1(\overline{\mathbb{R}^2_+}\setminus\overline{\Gamma_{\rm shock}^{\rm ext} \cup S_{12} \cup S_{32} })$

\begin{align}\label{eq:extension-of-varphi}
\varphi =
\begin{cases}
\max\{\varphi_5,\varphi_6\} &\,\,\, \text{in $\, \Omega_5\cup\Omega_6$}\,,\\
\bar{\varphi}  &\,\,\, \text{in $\, \mathbb{R}^2_+ \setminus (\Omega \cup \Omega_5 \cup \Omega_6)$}\,,
\end{cases}
\end{align}

where $\Omega_5$ and $\Omega_6$ are given in Definition <ref>, and $\bar{\varphi}$ is given
by (<ref>).

Equation (<ref>) is strictly elliptic in
$\cl{\Omega} \setminus ( \Gamma_{\rm{sonic}}^5 \cup \Gamma_{\rm{sonic}}^6)$, i.e.,
solution $\varphi$ satisfies

\begin{equation*}
|D\varphi(\bm{\xi})| < c(|D\varphi(\bm{\xi})|,\varphi(\bm{\xi})) \qquad
\text{for all $\bm{\xi} \in  \cl{\Omega}\setminus(\Gamma_{\rm{sonic}}^5 \cup \Gamma_{\rm{sonic}}^6)$}\,.
\end{equation*}

In $\Omega,$ solution $\varphi$ satisfies

\begin{align}
&D (\varphi_2 - \varphi) \cdot (\cos\theta,\sin\theta) \leq 0
\qquad\, \mbox{for all $\theta \in [\theta_{26},\theta_{25}]$}\,.
\label{DirecDerivMonotone-S25-S26}
\end{align}

From the definition of admissible solutions, we have

The requirement that \(\varphi\) is a solution of Problem <ref>
includes the condition that \(\varphi = \varphi_j\) and \(D\varphi = D\varphi_j\) on \(\Gamma_{\rm sonic}^j\)
for \(j = 5, 6,\) which becomes a one-point boundary condition in the case
\(\theta_{j-4} \in (\theta^{\rm s}, \theta^{\rm d}],\) according to the definition of \(\Gamma_{\rm sonic}^{j}\)
given by Definition <ref>.

Let \(\varphi\) be an admissible solution in the sense of Definition <ref>,
and let $\bm{\nu}$ be the unit normal vector on $\Gamma_{\rm shock}$ interior to $\Omega$.
Similarly to <cit.>, it can be shown that \(\Gamma_{\rm shock}\) is a transonic shock,
since \(\varphi\) satisfies

\begin{equation*}
D\varphi_2 \cdot \bm{\nu} > D\varphi \cdot \bm{\nu} > 0\,,
\quad
0 < \frac{D \varphi \cdot \bm{\nu}}{c(\abs{D\varphi},\varphi)} < 1 < D \varphi_2 \cdot \bm{\nu}
\qquad\,\, \text{on $\Gamma_{\rm shock}$}\,.
\end{equation*}

It follows from
Proposition <ref>(<ref>)
that $\varphi_j$ depends continuously on \(\theta_{j-4} \in [0,\theta^{\rm d}]\) for \(j = 5,6\).
Moreover, for any \(r > 0,\)

\begin{equation*}
\lim\limits_{\theta_{j-4} \to 0^+}\norm{\varphi_j - \varphi_0}_{C^{0,1}(B_r(\bm{0}) \cap \mathbb{R}^2_+)} = 0\,
\qquad\,\text{for $j=5,6$}\,,
\end{equation*}

i.e., the weak reflection state \((j)\) coincides with the normal reflection state $(0)$
when \(\theta_{j-4} = 0\). One can verify directly that the normal reflection solution
$\varphi_{\rm{norm}}\in C^{0,1}_{\rm loc}(\mathbb{R}^2_{+})$ as defined in (<ref>)
is an admissible solution corresponding to
$\btheta=\bm{0}$ in the sense of Definition

The first main theorem of this paper is to provide
the global existence of such admissible solutions in the sense of Definition <ref>.
The proof is given in <ref>.

Fix \(\gamma\geq1\) and  \(v_2 \in (v_{\min},0)\).
For any \(\btheta \in \Theta,\)
there exists an admissible solution \(\varphi\) of the four-shock interaction problem
corresponding to parameters \(\btheta\) in the sense of Definition <ref>.
Moreover, the global solution $\varphi$
converges to the normal reflection solution
$\varphi_{\rm{norm}}$ in $W_{\rm loc}^{1,1}(\mathbb{R}_+^2)$ as \(\btheta \in \Theta\) converges to \(\bm{0},\)
where \(\varphi_{\rm norm}\) is given by (<ref>).

The admissible solutions have additional regularity properties, given in Theorem <ref> below.
Moreover, the transonic shocks, as free boundaries of admissible solutions,
satisfy additional convexity properties, given in Theorem <ref> below.
The proofs of Theorems <ref>–<ref> are given
in <ref>–<ref>, respectively.

Fix $\gamma\geq1,$ $v_2 \in (v_{\min},0),$ and $\btheta \in \Theta$.
Let $\varphi$ be an admissible solution corresponding to parameters \(\btheta\) in the sense of Definition <ref>, with
transonic shock $\Gamma_{\rm{shock}}$.
Then the following properties hold:

*  $\Gamma_{\rm{shock}}$ is $C^{\infty}$ in its relative interior and

* I  When $\btheta \in [0,\theta^{\rm{s}})^2,$
$\varphi\in C^{\infty}(\overline{\Omega} \setminus ({\Gamma^5_{\rm{sonic}}}\cup{\Gamma^6_{\rm{sonic}}}) )\cap C^{1,1}(\overline{\Omega}),$ and
$\overline{S^{\rm seg}_{25}\cup\Gamma_{\rm{shock}}\cup S^{\rm seg}_{26}}$ is a $C^{2,\alpha}$–curve
for any $\alpha\in(0,1),$ including at points $P_2$ and $P_3$.

* II When $\btheta \in [\theta^{\rm{s}},\theta^{\rm{d}}) \times [0,\theta^{\rm{s}}),$
$\varphi\in C^{\infty}(\overline{\Omega}\setminus(\Gamma^6_{\rm{sonic}} \cup\{P_1\}) )
\cap C^{1,1}(\overline{\Omega}\setminus\{P_1\})\cap C^{1,\bar{\alpha}}(\overline{\Omega}),$ and
$\overline{S^{\rm seg}_{26}\cup\Gamma_{\rm{shock}}}$ is a $C^{1,\bar{\alpha}}$–curve for some $\bar{\alpha}\in(0,1)$.
Furthermore, $\overline{S^{\rm seg}_{26} \cup \Gamma_{\rm{shock}} } \setminus {\{P_{1}\}}$ is a $C^{2,\alpha}$–curve for any $\alpha\in(0,1),$ including at $P_3$.

When $\btheta \in [0,\theta^{\rm{s}}) \times [\theta^{\rm{s}},\theta^{\rm{d}}),$
$\varphi\in C^{\infty}(\overline{\Omega}\setminus (\Gamma^5_{\rm{sonic}}\cup\{P_4\}) )
\cap C^{1,1}(\overline{\Omega}\setminus\{P_4\})\cap C^{1,\bar{\alpha}}(\overline{\Omega}),$ and
$\overline{S^{\rm seg}_{25}\cup\Gamma_{\rm{shock}}}$ is a $C^{1,\bar{\alpha}}$–curve for some $\bar{\alpha}\in(0,1)$.
Furthermore, $\overline{S^{\rm seg}_{25}\cup\Gamma_{\rm{shock}}}\setminus{\{P_{4}\}}$
is a $C^{2,\alpha}$–curve for any $\alpha\in(0,1),$ including at $P_2$.

* III When $\btheta \in [\theta^{\rm{s}},\theta^{\rm{d}})^2,$
$\varphi\in C^{\infty}(\overline{\Omega}\setminus \{P_1,P_4\} )\cap C^{1,\bar{\alpha}}(\overline{\Omega}),$
and $\overline{\Gamma_{\rm{shock}}}$ is a $C^{1,\bar{\alpha}}$–curve for some $\bar{\alpha}\in(0,1)$.
Furthermore, $\overline{\Gamma_{\rm{shock}}}\setminus{\{P_{1},P_{4}\}}$ is a $C^{2,\alpha}$–curve for any $\alpha\in(0,1)$.

$\mathcal{U}\defeq \{\bm{\xi}\in\mathbb{R}^2_+ \,:\, \max\{\varphi_5(\bm{\xi}),\varphi_6(\bm{\xi})\}<\varphi_2(\bm{\xi})\}$.
For any constant $\sigma>0$ and \(j=5,6,\) define domain $\mathcal{U}^j_{\sigma}$ by

\begin{equation*}
\mathcal{U}^j_{\sigma}\defeq
\begin{cases}
\mathcal{U}\cap\big\{\bm{\xi}\in\mathbb{R}^2_+ \,:\, {\rm{dist}}(\bm{\xi},\Gamma_{\rm{sonic}}^{j}) < \sigma\big\}
\cap B_{c_j}(O_j)
&\quad \text{if $\theta_{j-4}\in [0,\theta^{\rm{s}})$}\,,\\
\varnothing\, &\quad \text{if $\theta_{j-4}\in[\theta^{\rm{s}},\theta^{\rm{d}})$}\,.
\end{cases}
\end{equation*}

Fix any point $\bm{\xi}_0\in({\Gamma^5_{\rm{sonic}}}\cup{\Gamma^6_{\rm{sonic}}})\setminus\{P_2,P_3\}$ and
denote $d\defeq \rm{dist}(\bm{\xi}_0,\Gamma_{\rm{shock}})$.
Then, for a sufficiently small constant $\varepsilon_0>0$ and for any $\alpha\in(0,1),$
there exists a constant $K<\infty$ depending on $(\gamma,v_2,\varepsilon_0,\alpha,d)$ and
$\|\varphi\|_{C^{1,1}(\Omega\cap(\mathcal{U}^5_{\varepsilon_0}\cup\,\mathcal{U}^6_{\varepsilon_0}))}$ such that

\begin{equation*}
\|\varphi\|_{2,\alpha,\overline{\Omega\cap B_{d/2}(\bm{\xi}_0)\cap(\mathcal{U}^5_{\varepsilon_0/2}\cup\, \mathcal{U}^6_{\varepsilon_0/2})}}\leq K\,;
\end{equation*}

*  For any $\bm{\xi}_0\in({\Gamma^5_{\rm{sonic}}}\cup{\Gamma^6_{\rm{sonic}}})\setminus\{P_2,P_3\},$

\begin{equation*}
\lim\limits_{ \bm{\xi}\to\bm{\xi}_0,\,\bm{\xi}\in \Omega}(D_{rr}\varphi-D_{rr}\max\{\varphi_5,\varphi_6\})(\bm{\xi})=\frac{1}{\gamma+1}\,,
\end{equation*}

where $r=|\bm{\xi}-O_j|$ for $\bm{\xi}$ near $\Gamma^j_{\rm{sonic}}$ for \(j=5,6\);

*  For \(i = 1, 2,\) whenever $\theta_{i}\in [0,\theta^{\rm{s}}),$
$\lim\limits_{ \bm{\xi}\to P_{i+1},\, \bm{\xi} \in \Omega }D^2\varphi (\bm{\xi})$ does not exist.

Fix \(\btheta \in \Theta\).
For any admissible solution
of the four-shock interaction problem in the sense of Definition <ref>, the transonic shock $\Gamma_{\rm{shock}}$ is uniformly convex
on closed subsets of its relative interior
in the sense described in Theorems <ref>–<ref>.
Furthermore, for any weak solution of (<ref>)–(<ref>)
in the sense of Definition <ref>,
satisfying also all the properties of Definition <ref> except (<ref>),
the transonic shock $\Gamma_{\rm{shock}}$ is a strictly convex graph if and only if condition (<ref>) holds.

§ UNIFORM ESTIMATES FOR ADMISSIBLE SOLUTIONS

To establish the first main theorem, Theorem <ref>,
we apply the Leray-Schauder degree
theorem to a suitable iteration map, similar to [2, 10].
To achieve this, in this section, we make several essential uniform estimates of
the admissible solutions.
In 4,  we construct a suitable iteration map between two function spaces with weighted Hölder norms,
where any fixed point of the iteration map is an admissible solution
in the sense of Definition <ref>.
The main results we establish in this section are the following:

* The strict directional monotonicity of \(\varphi_2-\varphi\), and the directional monotonicity of \(\varphi-\varphi_5\) and \(\varphi-\varphi_6\);

* The uniform positive lower bound of ${\rm dist}(\Gamma_{\rm{shock}},\partial B_1(O_2))$;

* The uniform estimate of the ellipticity of equation (<ref>) in \(\Omega\);

* The uniform estimates of admissible solutions in some suitable weighted Hölder norms.

Our problem shares many similarities with the Prandtl-Meyer shock reflection problem [2],
and the von Neumann shock reflection-diffraction problem [10].
Many of the following proofs are similar to those within [2, 10],
but have been adapted to the current setting with careful estimates.

§.§ Monotonicity and ellipticity

§.§.§ Directional monotonicity and uniform bounds

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
Let \(\varphi\) be an admissible solution corresponding to parameters \(\btheta \in \Theta\) in the sense of Definition <ref>,
and let the constant states $\varphi_2$  and $\varphi_j$, $j=5,6$, be given by (<ref>) and (<ref>) respectively.
We aim to establish the directional monotonicity properties for $\phi$ that satisfies

\begin{equation}\label{Eq4phi}
\qquad\, \text{in $\Omega$}\,,
\end{equation}

where $\phi$ can be taken as either $\varphi_2 - \varphi$ or $\varphi - \varphi_j$, $j=5,6$, and

\begin{equation}\label{Def4SonicSpeedInSec3}
\end{equation}

where $B$ is the Bernoulli constant given by (<ref>).

Fix \(\gamma\geq 1\) and \(v_2 \in (v_{\min},0)\).
For \(\btheta \in \Theta,\)
let \(\varphi\) be an admissible solution in the sense of Definition <ref> corresponding to parameters \(\btheta\).
Then \(\varphi\) satisfies that, for \(j = 5,6\),

\(\partial_{\bm{e}} (\varphi - \varphi_j)\) is not constant in \(\Omega\)
for any unit vector \(\bm{e} \in \mathbb{R}^2\);

For vector \(\bm{e}_{S_{2j}}\) given by (<ref>),

\begin{equation*}
\partial_{\bm{e}_{S_{2j}}}(\varphi_2 - \varphi) < 0 \qquad \text{in $\cl{\Omega} \setminus \Gamma_{\rm{sonic}}^j$}\,;
\end{equation*}

$\partial_{\bm{e}_{S_{2j}}} (\varphi - \varphi_j)\,, \, {-\partial_{\eta}}(\varphi - \varphi_j) \geq 0\;\; $ in  $\cl{\Omega}$ .

The proof of (<ref>) is given by a slight modification of <cit.>.
Indeed, for \(\phi \defeq \varphi - \varphi_5\), if \(\partial_{\bm{e}} \phi\) is constant in \(\Omega\),
then  \(\partial_{\bm{e}} \phi = 0 \) in \(\Omega\) since \(D\phi = \bm{0}\) at \(P_1\).
Furthermore, \(\partial_{\bm{e}} \phi = \bm{e}\cdot(u_6-u_5,0) \) at \(P_4\)
so that \(\bm{e}\) must be parallel to \((0,1)\), since \(u_5-u_6 > 0\) when \(\btheta\neq \bm{0}\).
Thus, \(\partial_\eta \phi \equiv 0\) in \(\Omega\) and hence
\(\partial_{\xi\eta} \phi = \partial_{\eta\eta} \phi \equiv 0\) in \(\Omega\).
By the strict ellipticity of equation (<ref>) in \(\Omega\),
\(\partial_{\xi\xi}\phi \equiv 0\) in \(\Omega\), which implies that
there exist constants \((u,v,k)\) such that \(\phi = u\xi + v\eta + k\) in \(\Omega\).
We note that \(v = 0\) since \(\partial_\eta \phi \equiv 0\).
Applying the boundary conditions: \(\phi = 0\) and \(D\phi =\bm{0}\) at \(P_1\),
we find that \(u = k = 0\), so that \(\phi \equiv 0 \) in \(\Omega\).
However, \(D\phi(P_4) = D\varphi_6(P_4) - D\varphi_5(P_4) = (u_6-u_5, 0) \neq \bm{0}\), which is a contradiction.
The proof is similar for \(\phi = \varphi - \varphi_6\).
This concludes the proof of (<ref>).

The proofs of (<ref>) and (<ref>) are similar to <cit.>
and <cit.>, respectively.

For any \(\btheta \in \cl{\Theta}\), define the following fan-shaped set by

\begin{equation*}
{\rm Cone}(\bm{e}_{S_{25}},\bm{e}_{S_{26}})\coloneqq
\big\{ r (\cos{\theta},\sin{\theta}) \,:\,  \theta_{26} \leq \theta \leq \theta_{25}, \, r \geq 0\big\}\,.
\end{equation*}

Let ${\rm Cone}^0(\bm{e}_{S_{25}},\bm{e}_{S_{26}})$ be the interior of ${\rm Cone}(\bm{e}_{S_{25}},\bm{e}_{S_{26}})$.
It follows from Lemma <ref> that,
if \(\varphi\) is an admissible solution corresponding to \(\btheta\in\Theta\), then \(\varphi\) satisfies

\begin{align}
\partial_{\bm{e}} (\varphi_2 - \varphi ) < 0
\qquad
\text{in $\cl{\Omega}\, $ for any $\bm{e} \in {\rm Cone}^0 (\bm{e}_{S_{25}},\bm{e}_{S_{26}})$}\,,
\label{nStrictDirectMonotPropty}
\end{align}

and ${-}\bm{\nu}(P) \in \big\{ (\cos\theta, \sin\theta) \,:\,
\theta_{25} - \tfrac{\pi}{2} < \theta < \theta_{26} + \tfrac{\pi}{2} \big\}$
for all $P \in \Gamma_{\rm{shock}}$ ,
where $\bm{\nu}$ is the unit normal vector on \(\Gamma_{\rm{shock}}\) pointing into the interior of \(\Omega\).

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For \(\btheta \in \Theta,\) let $\varphi$ be an admissible solution in the sense of Definition <ref> with parameters $\btheta$.
Then there exists a $C^1$–function
\(f_{\rm sh}\) such that

\begin{equation*}
\Gamma_{\rm{shock}}=\{ \bm{\xi}\in\mathbb{R}^2_+ \,:\,  \eta = f_{\rm{sh}}(\xi), \, \xi^{P_3} < \xi < \xi^{P_2} \}\,,
\end{equation*}

and $f'_{\rm sh}(\xi)\in(\tan\theta_{25},\tan\theta_{26})$ for $\xi\in(\xi^{P_3},\xi^{P_2})$
with $f'_{\rm{sh}}(\xi^{P_{j-3}})=\tan\theta_{2j},$ for $j=5,6$.

By Lemma <ref>, $\bm{e}_{\eta} = (0,1) \in {\rm Cone}^0(\bm{e}_{S_{25}},\bm{e}_{S_{26}})$.
From the strict directional monotonicity (<ref>), we see that
on $\overline{\Gamma_{\rm{shock}}}$,
which implies the existence of a $C^1$–function $f_{\rm{sh}}$ that solves the implicit relation $\varphi_{2}(\xi,f_{\rm sh}(\xi))=\varphi(\xi,f_{\rm sh}(\xi))$.
Taking the derivative, we have

\begin{equation*}
\end{equation*}

From Definition <ref>(<ref>), we obtain that
and $f'_{\rm{sh}}(\xi^{P_3})=\tan\theta_{26}$.
Denoting by $\bm{\nu}(P)$ the unit normal vector on \(\Gamma_{\rm shock}\) at a point $P \in \Gamma_{\rm shock}$,
interior to \(\Omega\), we have

\begin{equation*}
\bm{\nu}(P) =\frac{D(\varphi_2-\varphi)(P)}{|D(\varphi_2-\varphi)(P)|}=\frac{(f'_{\rm{sh}}(\xi),-1)}{\sqrt{1+\big(f'_{\rm{sh}}(\xi)\big)^2}}\,.
\end{equation*}

Combining the above with property (<ref>) in Lemma <ref>, for any \((a_1,a_2) \in [0,\infty)^2\), \((a_1,a_2)\neq (0,0)\),

\begin{equation*}\begin{aligned}
&a_1\big(f'_{\rm{sh}}(\xi)\cos\theta_{25}-\sin\theta_{25}\big)+a_2\big(f'_{\rm{sh}}(\xi)\cos\theta_{26}-\sin\theta_{26}\big)  \\
&\;\;= \sqrt{1+\big(f'_{\rm{sh}}(\xi)\big)^2} \; \bm{\nu}(P)  \cdot(a_1\bm{e}_{S_{25}}+a_2\bm{e}_{S_{26}})<0
\qquad\,\,\mbox{for any $\xi\in(\xi^{P_3},\xi^{P_2})$}\,.
\end{aligned}
\end{equation*}

Choosing $(a_1,a_2)$ as $(1,0)$ and $(0,1)$ in turn, we obtain that
$f'_{\rm{sh}}(\xi)>\tan\theta_{25}$ and $f'_{\rm{sh}}(\xi)<\tan\theta_{26}$, respectively, where we have used $\theta_{25}\in(\tfrac{\pi}{2},\pi]$ and $\theta_{26}\in [0,\frac{\pi}{2})$.

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For any $\btheta \in \Theta,$ let $\varphi$ be an admissible solution corresponding to
parameters $\btheta \in \Theta$ in the sense of Definition <ref>,
and let $\Omega$ be the pseudo-subsonic region of \(\varphi\).
Then there exists a constant $C_{\rm ub}>0$ depending only on $(\gamma,v_2)$ such that the following properties hold:

\begin{align}
\label{UB-01}
&\cl{\Omega} \subseteq B_{C_{\rm ub}}(\bm{0})\,, \\
\label{UB-02}
&\|\varphi\|_{C^{0,1}(\overline{\Omega})}\leq C_{\rm ub}\,, \\
\label{UB-03}
&\rho^*(\gamma)\leq\rho\leq C_{\rm ub} \quad \text{in $\Omega$}\,, \qquad 1<\rho\leq C_{\rm ub} \quad \text{on $\Gamma_{\rm{shock}}$}\,,
\end{align}

where $\rho^*(\gamma) \defeq (\frac{2}{\gamma+1})^{\frac{1}{\gamma-1}}$ when $\gamma>1,$ and $\rho^*(1) \defeq e^{-\tfrac{1}{2}}=\lim\limits_{\gamma\to1+}\rho^*(\gamma)$.

We divide the proof into three steps.

1. To prove (<ref>), we use the expression of $\Gamma_{\rm{shock}}$ from Lemma <ref>.
In particular,
it follows from
from Definition <ref>(<ref>) and Lemma <ref>
$0 < f_{\rm{sh}}(\xi) \leq \min\{\xi \tan \theta_{25} + a_{25}, \xi \tan \theta_{26} + a_{26}\}$
whenever \(\xi^{P_3} < \xi < \xi^{P_2}\), which implies that

\begin{equation*}
\Omega \subseteq \big\{\bm{\xi} \,:\, u_6-c_6 < \xi < u_5+c_5 , \,
0 < \eta \leq \max\{a_{25}, a_{26}\}\big\}\,.
\end{equation*}

From (<ref>) and Proposition <ref>(<ref>)
and Definition <ref>,
we know that $(\rho_5,u_5, a_{25})$
depend continuously on $\theta_1\in[0,\theta^{\rm{d}}]$,
while $(\rho_6,u_6,a_{26})$
depend continuously on $\theta_2\in[0,\theta^{\rm{d}}]$,
from which (<ref>) follows directly.

2. From Definition <ref>(<ref>), we see that
$\inf_{\overline{\Omega}} \max \{\varphi_5,\varphi_6\} \leq \varphi \leq \sup_{\overline{\Omega}} \varphi_2$.
By (<ref>) and the expressions of $(\varphi_2,\varphi_5,\varphi_6)$ given in (<ref>)
and (<ref>), there exists a constant $C_1>0$ depending only on $(\gamma,v_2)$ such that
$-C_1 \leq \min_{\cl{\Omega}}\max\{\varphi_5,\varphi_6\} \leq \max_{\cl{\Omega}} \varphi_2 \leq C_1$,
which implies that $\max\limits_{\cl{\Omega}} |\varphi| \leq C_1$.
Then the uniform bound (<ref>) follows from

\begin{equation*}
\end{equation*}

where we have used Definition <ref>(<ref>) and (<ref>).

3. The upper bounds in (<ref>) follow directly from the expression of density (<ref>) along with (<ref>).
For the lower bound in \(\Omega\), we combine the Bernoulli law (<ref>) with Definition <ref>(<ref>) to obtain

\begin{equation*}\begin{aligned}
\frac12|D\varphi|^2 + h(\rho)
= \frac12|D\varphi_2|^2 + (\varphi_2-\varphi) + h(\rho_2)\geq \frac12 |D\varphi_2|^2 \geq 0\,.
\end{aligned} \end{equation*}

Then, from Definition <ref>(<ref>), we have

\begin{equation*}
\frac{1}{\gamma-1} \big( \frac12 (\gamma+1) \rho^{\gamma-1} - 1 \big) = \frac12c^2 + h(\rho) \geq \frac12|D\varphi|^2+h(\rho)\geq0\,,
\end{equation*}

so that the lower bound of $\rho$ in $\Omega$ is obtained.
For the lower bound of $\rho$ on \(\Gamma_{\rm shock}\),
we combine Problem <ref>(<ref>) and Remark <ref>(<ref>)
to obtain

ρ= Dφ_2·ν/Dφ·ν > 1      .

§.§.§ Uniform positive lower bound of \({\rm dist}(\Gamma_{\rm{shock}}, \partial B_1 (O_2))\)

We now obtain a uniform estimate for the positive lower bound of
\({\rm{dist}}(\Gamma_{\rm{shock}}, \partial B_1(O_2))\) for any admissible solution.
This allows us to make the uniform estimate for the ellipticity of equation (<ref>) within the pseudo-subsonic domain of admissible solutions.

Fix \(\gamma \geq 1\) and \(v_2 \in (\vmin,0)\).
There exists a constant \(C_{\rm sh}>0\) depending only on \((\gamma, v_2)\) such that any admissible solution corresponding to parameters \(\btheta \in \Theta \) satisfies

\begin{equation}\label{Eq:Result-Prop-3-4}
{\rm{dist}} \big( \Gamma_{\rm{shock}}, \, \partial B_1(O_2) \big) \geq C_{\rm sh}^{-1}\,.
\end{equation}

The proof of (<ref>) follows the same argument
as for <cit.> except the case that \( v_2 \leq -1\).
In the following, we focus on the case that \( v_2 \leq -1\) and give the proof
in three steps.

1. We first rewrite equation (<ref>) as

\begin{equation*}\label{Newnotation4Eq}
\mbox{div}\mathcal{A}(D\varphi,\varphi)+\mathcal{B}(D\varphi,\varphi) = 0\,,
\end{equation*}

$\mathcal{A}(\bm{p},z)\defeq \rho(|\bm{p}|,z)\bm{p}$ and
$\mathcal{B}(|\bm{p}|,z)\defeq 2\rho(|\bm{p}|,z)$
with $\bm{p}=(p_1,p_2)\in\mathbb{R}^2$, $z\in\mathbb{R}$, and

\begin{equation}\label{Eq:DefRhopzSec302}
\rho(|\bm{p}|,z)=
\begin{cases}
\big(1+(\gamma-1)(\frac{1}{2}v_2^2-\frac{1}{2}|\bm{p}|^2-z)\big)^{\frac{1}{\gamma-1}}
&\quad\text{if} \; \gamma>1\,,\\
\,\exp(\frac{1}{2}v_2^2-\frac{1}{2}|\bm{p}|^2-z)  &\quad \text{if} \;\gamma = 1\,.\\
\end{cases}
\end{equation}

We denote the sonic speed as $c(|\bm{p}|,z) \defeq \rho^{(\gamma-1)/2}$ for $\gamma\geq1$.
For a constant $R>1$, define

\begin{equation}\label{KRSet}
\mathcal{K}_R \defeq
\Big\{(\bm{p},z) \in \mathbb{R}^2 \times \mathbb{R} \,:\, |\bm{p}| + |z| \leq R, \,
\rho(|\bm{p}|,z) \geq R^{-1}, \,\frac{|\bm{p}|^2}{c^2(|\bm{p}|,z)} \leq 1-R^{-1}
\Big\}\,.
\end{equation}

For each $R>1$, there exists a constant $\lambda_R>0$ depending only on $(\gamma,v_2,R)$ such that,
for any $\bm{\kappa}=(\kappa_1,\kappa_2)\in\mathbb{R}^2$,

\begin{equation*}
\sum_{i,j=1}^2\partial_{p_j}\mathcal{A}_i(\bm{p},z)\kappa_i\kappa_j\geq\lambda_R|\bm{\kappa}|^2
\qquad\, \text{for any $(\bm{p},z)\in\mathcal{K}_R$}\,.
\end{equation*}

The next two lemmas follow directly from <cit.>.

For $R>2,$ let $\mathcal{K}_R$ be given by (<ref>).
Then there exist functions $(\tilde{\mathcal{A}},\tilde{\mathcal{B}})(\bm{p},z)$ in $\mathbb{R}^2\times\mathbb{R}$ satisfying the following properties:

For any $(\bm{p}_0,z_0)\in\mathcal{K}_R,$
= (\mathcal{A},\mathcal{B})(\bm{p},z)$
for all $(\bm{p},z)\in B_{\varepsilon}((\bm{p}_0,z_0))$;

For any $(\bm{p},z) \in\mathbb{R}^2\times\mathbb{R}$ and $\bm{\kappa}=(\kappa_1,\kappa_2)\in\mathbb{R}^2,$

$|\tilde{\mathcal{B}}(\bm{p},z)|\leq C_0$
$|D^m_{(\bm{p},z)}(\tilde{\mathcal{A}},\tilde{\mathcal{B}})(\bm{p},z)|\leq C_m$
in $\mathbb{R}^2\times\mathbb{R}$ for each $m=1,2,\cdots,$

where the positive constants
$\varepsilon$ and $\lambda_{R}$ depend only on $(\gamma,v_2,R)$,
and $C_m$ depends on $(\gamma,v_2,R,m)$.

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For any given constants $\alpha\in(0,1),$ $m\in\mathbb{N},$ and $r>0,$ there exist constants $C,C_m>0$
depending only on $(\gamma,v_2,\alpha,r),$ with $C_m$ depending additionally on $m,$
such that any admissible solution $\varphi$ corresponding to parameters
$\btheta\in\Theta$ satisfies the following estimates:

For any $B_{4r}(P)\subseteq\Omega,$

\begin{align*}
\|\varphi\|_{2,\alpha,\overline{B_{2r}(P)}} \leq C\,, \qquad
\|\varphi\|_{m,\overline{B_{r}(P)}} \leq C_m\,;
\end{align*}

If $P\in\Gamma_{\rm{sym}},$ and $B_{4r}(P)\cap\Omega$ is the half-ball $B^+_{4r}(P) \defeq B_{4r}(P)\cap\{\eta>0\},$ then

φ_2,α,B_2r(P)∩Ω ≤C ,     φ_m,B_r(P)∩Ω ≤C_m .

Fix $\varepsilon>0$ small enough such that $2\xi^{P_3}<\xi^{P_0}$ for all $\theta_2\in[0,\varepsilon)$,
where $\xi^{P_0} < 0$ is the $\xi$–coordinate of $P_0\defeq\lim\limits_{\theta_2\to0^+}P_3$, which depends only on $(\gamma,v_2)$.
We define

\begin{equation}\label{Def4dsep}
d_{\rm sep}\defeq \min{\big\{ {-}\frac{\xi^{P_0}}{2}, \, \inf\limits_{\theta_2\in[\varepsilon,\theta^{\rm d}]}|u_6|\big\}}\,.
\end{equation}

It is direct to verify that \(d_{\rm sep} > 0\) by using \(u_6 = v_2\tan\theta_{26}\)
and property (<ref>), and $d_{\rm sep}$ depends only on $(\gamma,v_2)$.
Then, for any $\btheta\in\Theta$,  ${\rm dist}(P_2,P_3)\geq 2 d_{\rm sep}$.
We also define

\begin{equation}\label{eq:defOfr1}
r_1 \defeq \frac{1}{2} \inf_{\btheta \in \Theta}
\big\{ | D\varphi_5 (P_0^1 ) | , | D\varphi_6 (P_0^2) | \big\}\,.
\end{equation}

By (<ref>) and Lemma <ref>,
it is direct to verify that $r_1>0$ depends only on $(\gamma,v_2)$.

For any $r\in(0,r_1)$ and for $i=1,2$, we define two sets:

\begin{equation*}
\begin{aligned}
&   \mathscr{A}\defeq \big\{\theta_i\in(0,\theta^{\rm d}] \,:\, |P_0^i-P_{i+1}|
\geq \frac{r}{20}\big\}\cup\{0\}\,,\quad  \mathscr{B}\defeq \big\{\theta_i\in[0,\theta^{\rm d}] \,:\,
|P_{3i-2}-P_{i+1}| \geq \frac{r}{20}\big\}\,.
\end{aligned}
\end{equation*}

From the continuous dependence of $P_0^i,P_{3i-2}$, and $P_{i+1}$ on $\theta_i\in(0,\theta^{\rm d}]$,
we know that $\mathscr{A}$ and $\mathscr{B}$ both are closed sets.
Then there exists a constant $C_1>0$ such that

\begin{equation*}
{\rm dist}(P_{i+1},\Gamma_{\rm{sym}})\geq\frac{2}{C_1}
\qquad \text{for all $\theta_i\in \mathscr{A}\cup\mathscr{B}$}\,.
\end{equation*}

If ${\rm dist}(P_{i+1},\Gamma_{\rm{sym}})\leq\frac{1}{C_1}$,
then $\theta_i\notin \mathscr{A}\cup \mathscr{B}$, which implies that

\begin{equation*}
\max\{|P_0^i-P_{i+1}|, |P_{3i-2}-P_{i+1}|\} < \frac{r}{20}\,.
\end{equation*}

We now seek a constant $C_r > 0$ depending only on $(\gamma,v_2,r)$ such that,
for any admissible solution of Problem <ref> corresponding to parameters $\btheta\in\Theta$,

\begin{equation}\label{Eq:Lem942a}
\sup\limits_{P\in\Gamma_{\rm{shock}}\cap B_{r/2}(P_{3i-2})}{\rm{dist}}(P,\Gamma_{\rm{sym}})
>C_r^{-1}\qquad  \text{if $|P_{3i-2}-P_{i+1}|\leq\frac{r}{10}$}\,.
\end{equation}

To obtain the above result, we follow the proof as in <cit.>,
in which we need to take a limit of parameters $\btheta^{(n)}$.

In order to take the limit, we use a
compactness lemma below.
For concreteness, in the following, we use the notation defined in Definition <ref>,
and append superscripts $(n)$ and $*$ to indicate that the corresponding quantities are related
to parameters $\btheta^{(n)}$ and $\btheta^{*}$, respectively.

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
Let $\big\{ \btheta^{(n)}\big\}_{n \in \mathbb{N}} \subseteq \Theta$ be any sequence of parameters satisfying
$\, \lim_{n\to\infty}\btheta^{(n)} = \btheta^{*}$
for some $\btheta^{*} \in \cl{\Theta}$.
For each \(n\in\mathbb{N},\) let
$\varphi^{(n)}$ be an admissible solution corresponding to parameters $\btheta^{(n)}$.
Then there exists a subsequence $\{\varphi^{(n_j)}\}_{j\in\mathbb{N}}$ converging uniformly on
any compact subset of $\mathbb{R}^2_+$ to a function $\varphi^*\in C^{0,1}_{\rm loc}(\overline{\mathbb{R}^2_+})$.
Moreover, $\varphi^*$ is a weak solution of equation (<ref>)
in $\mathbb{R}^2_+$ and satisfies the following properties:

For the corner points $P_m,$ $m=1,2,3,4,$ and the reflection points $P_0^i,$ $i=1,2,$

\begin{eqnarray*}
\lim\limits_{j\to\infty}P_m^{(n_j)}=P_m^*\,, \qquad
\lim\limits_{j\to\infty}P_0^{i,(n_j)}=P_0^{i,*}\,.
\end{eqnarray*}

Note that $\xi^{P_0^{1,*}} = \infty$ in the case that $\theta^*_1=0,$ and $\xi^{P_0^{2,*}}=-\infty$
in the case that $\theta^*_2=0$.

Let $\eta=f_{2l}^{(n_j)}(\xi),$ $l=5,6,$ and $\eta=f^{(n_j)}_{\rm{sh}}(\xi)$ be the expressions
of $S^{(n_j)}_{2l}$ and $\Gamma^{(n_j)}_{\rm shock}$
from Definition <ref> and Lemma <ref>, respectively.
Extend $f_{\rm{sh}}^{(n_j)}$ by

\begin{equation*}\label{Express4ShSeq}
f^{(n_j)}_{\rm{sh}}(\xi) =
\begin{cases}
\, f_{25}^{(n_j)}(\xi) &\quad \text{for $\xi^{P^{(n_j)}_2}< \xi < \xi^{P_0^{1,(n_j)}}$}\,,\\
\, f_{26}^{(n_j)}(\xi)  &\quad \text{for $\xi^{P_0^{2,(n_j)}} < \xi < \xi^{P_3^{(n_j)}}$}\,.
\end{cases}
\end{equation*}

$\{f^{(n_j)}_{\rm{sh}}\}_{j\in\mathbb{N}}$ converges uniformly
to a function $f^{*}_{\rm{sh}}\in C^{0,1}([\xi^{P_0^{2,*}},\xi^{P_0^{1,*}}])$.

Let smooth functions $\eta = g^{(n_j)}_{l,\rm{so}}(\xi)$
be the expressions of $\Gamma^{l,(n_j)}_{\rm{sonic}}$ for $l=5,6$.
Then $\{g^{(n_j)}_{5,\rm{so}}\}_{j\in\mathbb{N}}$ converges pointwise
to $g^{*}_{5,\rm{so}}$ on $(\xi^{P^{*}_2},\xi^{P^{*}_1}),$ and $\{g^{(n_j)}_{6,\rm{so}}\}_{j\in\mathbb{N}}$
converges pointwise to $g^{*}_{6,\rm{so}}$ on $(\xi^{P^{*}_4},\xi^{P^{*}_3})$.

$\widehat{\Omega^*}\defeq \big\{\bm{\xi}\in[\xi^{P^*_4},\xi^{P^*_1}]\times\overline{\mathbb{R}_+}\,:\,
0\leq\eta\leq f^*_{\rm{bd}}(\xi)\big\},$
where $f^*_{\rm{bd}}(\cdot)$ is given by

\begin{equation*}
f^*_{\rm{bd}}(\xi) =
\begin{cases}
\, g^{*}_{6,\rm{so}}(\xi)  &\quad \text{for $\xi^{P^*_4}\leq\xi\leq\xi^{P^*_3}$}\,,\\
\, f^*_{\rm{sh}}(\xi) \qquad  &\quad \text{for $\xi^{P^*_3}<\xi\leq\xi^{P_2}$}\,,\\
\, g^*_{5,\rm{so}}(\xi) \qquad  &\quad \text{for $\xi^{P^*_2}<\xi\leq\xi^{P^*_1}$}\,.
\end{cases}
\end{equation*}

Denote by $\Omega^*$ the interior of $\widehat{\Omega^*},$
$\Gamma^*_{\rm{shock}} \defeq \{(\xi,f^*_{\rm{sh}}(\xi)) \,:\, \xi\in(\xi^{P^*_3},\xi^{P^*_2})\},$
and $\Gamma^*_{\rm{sym}}\defeq \{(\xi,0) \,:\, \xi\in(\xi^{P^*_4},\xi^{P^*_1})\},$
and denote by $\Gamma^{*,0}_{\rm{sym}}$ the relative interior of $\Gamma^*_{\rm{sym}}\setminus\Gamma^*_{\rm{shock}}$.
Then $\varphi^* \in C^\infty(\Omega^*\cup\Gamma^{*,0}_{\rm{sym}}),$ and

$\lim\limits_{j\to\infty}\|\varphi^{(n_j)}-\varphi^*\|_{C^2(K)}=0\,$ for any $K \Subset \Omega^* \cup \Gamma^{*,0}_{\rm{sym}}$,

$D (\varphi_2 - \varphi^*) \cdot (\cos\theta,\sin\theta) \leq 0\,$ for all $\theta \in (\theta^*_{26},\theta^*_{25})$,

Equation (<ref>) is strictly elliptic in $\Omega^*\cup\Gamma^{*,0}_{\rm{sym}}$,

In $\mathbb{R}^2_+\setminus\Omega^*,$ $\varphi^*$ is equal to the constant states $\bar{\varphi}^*,$ $\varphi^*_5,$
and $\varphi^*_6$ in their respective domains as given
in Definition <ref>(<ref>),
where $\bar{\varphi}^*$ is given by (<ref>) with $\btheta = \btheta^*$.

$\Gamma_{\rm{sym}}^{*,0}=\{\varphi_2>\varphi^*\}\cap\Gamma_{\rm{sym}}^{*}$ is dense in $\Gamma_{\rm{sym}}^{*}$.

Using Lemmas <ref> and <ref>,
properties (<ref>)–(<ref>) in Lemma <ref>
can be proved by
similar arguments
as for <cit.>.

For property (<ref>), if it is not true,
we can choose $\xi_1,\xi_2\in(\xi^{P_3^*},\xi^{P^*_2})$ such that
$\{(\xi,0) \,:\, \xi_1<\xi<\xi_2\}\subseteq\{\varphi_2\leq\varphi^*\}\cap\Gamma_{\rm{sym}}^{*}$.
By Definition <ref>(<ref>),
$\varphi^{(n_j)}\leq\varphi_2$ in $\Omega^{(n_j)}$ so that
$\varphi^{*}\leq\varphi_2$ in $\overline{\Omega^{*}}$.
Combining with property (iv.4), we obtain that $\varphi^*(\bm{\xi})=\varphi_2(\bm{\xi})$ on $\{\bm{\xi}\in\overline{\mathbb{R}_+^2}: \xi_1<\xi_2\}$.
We take any non-negative function $\zeta\in C^{\infty}_{\rm c}(\overline{\mathbb{R}^2_+})$
satisfying that ${\rm supp} \, \zeta\subseteq \{\bm{\xi}\in\overline{\mathbb{R}^2_+} \,:\, \xi_1 < \xi < \xi_2 \}$
and $\zeta > 0$ on some subset of $\{(\xi,0) \,:\, \xi_1 < \xi < \xi_2 \}$ with positive measure.

\begin{align*}
&\int_{\mathbb{R}^2_+} (\rho(|D\varphi^*|,\varphi^*) D\varphi^* \cdot D\zeta(\bm{\xi}) - 2\rho(|D\varphi^*|,\varphi^*) \zeta(\bm{\xi})) \, {\rm d}\bm{\xi}\\
&\;\;= \int_{\mathbb{R}^2_+} ( \rho_2 D\varphi_2 \cdot D\zeta(\bm{\xi}) - 2 \rho_2 \zeta(\bm{\xi})) \, {\rm d} \bm{\xi}
= -v_2 \int_{\xi_1}^{\xi_2} \zeta(\xi,0) \, {\rm d} \xi > 0\,,
\end{align*}

which contradicts the fact that $\varphi^*$ is a weak solution of
equation (<ref>) in $\mathbb{R}^2_+$.
Then Lemma <ref>(<ref>) is verified.

We prove (<ref>) by
contradiction. If it is not true, we can choose a sequence of parameters $\{\btheta^{(n)}\}_{n\in\mathbb{N}}$ such that

\begin{equation*}
\sup\limits_{P\in\Gamma^{(n)}_{\rm{shock}}\cap B_{r/2}(P^{(n)}_{3i-2})}{\rm{dist}}(P,\Gamma^{(n)}_{\rm{sym}})\leq\frac{1}{n} \qquad
\text{if $|P^{(n)}_{3i-2}-P^{(n)}_{i+1}|\leq\frac{r}{10}$}\,.
\end{equation*}

Then, by Lemma <ref>, we can further choose
a subsequence $\{\btheta^{(n_j)}\}_{j\in\mathbb{N}}$ and take a limit
to some $\btheta^{*}\in\overline{\Theta}$, which provides a contradiction
to Lemma <ref>(<ref>).

From (<ref>), we can choose $\hat{Q}_i\in\overline{\Gamma_{\rm{shock}}\cap B_{r/2}(P_{3i-2})}$
with ${\rm dist}(\hat{Q}_i,\Gamma_{\rm{sym}})\geq C_r^{-1}$,
so that
$\hat{Q}_i\in \overline{\Gamma_{\rm shock} \cap B_{3r/4}(P_0^i)}$.
For $i = 1,2$, we take $Q_i=P_{i+1}\in\overline{\Gamma_{\rm shock}}$
if ${\rm dist}(P_{i+1},\Gamma_{\rm sym})> C_1^{-1}$;
otherwise, we take $Q_i=\hat{Q}_{i}\in\overline{\Gamma_{\rm shock}\cap B_{3r/4}(P_0^i)}$ as chosen above.
Then it follows that
\begin{equation*}
\min\{{\rm dist}(Q_1,Q_2), \, {\rm dist}(Q_1,\Gamma_{\rm sym}),\, {\rm dist}(Q_2,\Gamma_{\rm sym})\}
\geq \min\{d_{\rm sep}, \, C_1^{-1}, \, C_{r}^{-1}\} \eqdef a\,.
\end{equation*}

Combining with Lemma <ref> and following a similar proof
as for <cit.>, we obtain that,
$a>0$ defined above, there exists a constant $C_2>0$ depending only on $(\gamma,v_2,a)$
such that, for any admissible solution
corresponding to parameters $\btheta\in\Theta$,

\begin{equation*}
{\rm dist}(\Gamma_{\rm shock}[Q_1,Q_2], \,\Gamma_{\rm sym})\geq C_2^{-1}\,,
\end{equation*}

where $\Gamma_{\rm shock}[Q_1,Q_2]$ represents the segment on $\Gamma_{\rm shock}$ between points $Q_1$ and $Q_2$.

\begin{equation}\label{Eq:ResultInStep3}
{\rm dist}(\Gamma_{\rm shock} \setminus ( B_r (P_0^1) \cup B_r (P_0^2) ) , \, \Gamma_{\rm{sym}} ) > C_2^{-1}\,.
\end{equation}

3. Since \( v_2 \leq -1\), \(B_1(O_2) \subseteq \mathbb{R}\times(-\infty,0)\).
By Lemma <ref> and property (<ref>) of Proposition <ref>,
\(d_{\rm ref}\defeq |D\varphi_2(P_0^1|_{\theta_1 = \theta^{\rm d}})| - 1 > 0\) depending only on \((\gamma, v_2)\) satisfies

\begin{align*}
{\rm dist}(P_0^1, \,B_1(O_2))
= |D\varphi_2(P_0^1)| - 1 \geq d_{\rm ref}\,.
\end{align*}

By symmetry, we also see that ${\rm dist}(P_0^2, \, B_1(O_2)) \geq d_{\rm ref}$.
Denote \(\bar{r} \defeq \frac12 \min \{r_1, d_{\rm ref}\} > 0\)
with \(r_1\) defined by (<ref>). Then

\begin{equation*}
{\rm dist}(\Gamma_{\rm shock} \cap (B_{\bar{r}}(P_0^1) \cup B_{\bar{r}}(P_0^2)) , \, B_1(O_2) ) \geq \bar{r}\,.
\end{equation*}

Meanwhile, from the argument in Step 2, there exists a constant \(C_2>0\)
depending only on \((\gamma, v_2, \bar{r})\) such that (<ref>) holds
with $r$ replaced by $\bar{r}$.
Thus, for any admissible solution corresponding to parameters \(\btheta \in \Theta\), we have

\begin{align*}
{\rm dist}(\Gamma_{\rm shock}, \, B_1(O_2) ) \geq \min\{ \bar{r}, \, C_2^{-1}\} \eqdef C_{\rm sh}^{-1}\,.
\end{align*}

This completes the proof of (<ref>).

§.§.§ Uniform estimate for the ellipticity of equation (<ref>)

We now show a uniform estimate for the ellipticity of equation (<ref>),
which may be degenerate near $\Gamma_{\rm{sonic}}^{5}\cup\Gamma_{\rm{sonic}}^{6}$.
Fix a function \(h^\ast \in C^\infty(\mathbb{R})\) such that

\begin{equation*}
h^\ast (s) =
\begin{cases}
\, s &\quad \text{for $s\leq \frac12$}\,, \\[1mm]
\, 1 &\quad \text{for $s\geq1$}\,,
\end{cases}
\end{equation*}

satisfying $0 \leq (h^\ast)'(\cdot)\leq 2$ on $\mathbb{R}$.
For each \( \btheta \in \cl{\Theta}\), define

\begin{equation}
\label{Eq:Sec3-Def4hatcj}
\hat{c}_j \defeq \min{\{ c_{j}, \, \abs{D\varphi_{j} (P_0^{j-4})} \}}\,,
\quad
q_j  \defeq {\rm dist}(O_j, \, S_{2j}) \qquad\, \mbox{for $j=5,6$}\,.
\end{equation}

By Lemma <ref> and <cit.>,
we see that \(\hat{c}_j > q_j\) for every \(\btheta \in \overline{\Theta}\), so that
there exists a constant \(\hat{\delta}>0\) depending only on \((\gamma,v_2)\) such that

\begin{equation}\label{Eq:DistO5P01minusO5S25}
\inf_{\btheta \in \cl{\Theta}}\min\{\hat{c}_5 - q_5, \, \hat{c}_6 - q_6\} \geq \hat{\delta} \,.
\end{equation}

For $j=5,6$, define \(g^\ast_j:\mathbb{R}^2 \to \mathbb{R}_+\) by

\begin{equation*}
g^\ast_j(\bm{\xi}) \defeq \frac{1}{2} (\hat{c}_j - q_j)
h^\ast (\frac{{\rm{dist}}(\bm{\xi}, \, \partial B_{\hat{c}_j}(O_j))}{\hat{c}_j - q_j})
\end{equation*}

We choose a function \(\chi\in C^{\infty}(\mathbb{R})\) such that

\begin{equation*}
\chi(\xi) =
\begin{cases}
\,1  & \quad \text{for} \;\xi \leq -\tfrac{d_{\rm sep}}{4}\,, \\
\,0  &  \quad \text{for} \;\xi \geq \tfrac{ d_{\rm sep}}{4}\,,
\end{cases}
\end{equation*}

satisfying  $-\frac{4}{d_{\rm sep}} \leq\chi'(\cdot)\leq 0$ on $\mathbb{R}$,
where $d_{\rm sep}>0$ is given by (<ref>) depending only on \((\gamma,v_2)\).
Finally, define a function \(g:\mathbb{R}^2 \to \mathbb{R}_+\) by

\begin{equation}  \label{eq:g_theta1theta2}
g(\bm{\xi}) \defeq \chi(\xi) g_6^\ast(\bm{\xi})  + (1 - \chi(\xi))g_5^\ast(\bm{\xi})\,.
\end{equation}

By Definition <ref> and Lemma <ref>,
there exist constants \(d>0\) and \(C>1\) depending only on \((\gamma, v_2)\) such that,
if \(\varphi\) is an admissible solution corresponding to \(\btheta\in\Theta\)
with pseudo-subsonic region \(\Omega\),
then \(g\) given by (<ref>) satisfies the following properties:

For \(\bm{\xi}\in\Omega\) satisfying \({\rm{dist}}(\bm{\xi},\Gamma^j_{\rm{sonic}}) < d\),
$C^{-1} {\rm{dist}}_{j}(\bm{\xi}, \,\Gamma_{\rm{sonic}}^j) \leq
g(\bm{\xi}) \leq
C\,{\rm{dist}}_{j}(\bm{\xi}, \,\Gamma_{\rm{sonic}}^j)$,
where ${\rm{dist}}_{j}(\bm{\xi}, \,\Gamma_{\rm{sonic}}^j)
\defeq {\rm{dist}}(\bm{\xi}, \,\Gamma_{\rm{sonic}}^j) + ( c_j - \hat{c}_j
)$ for $j=5,6$.

For each \(\varepsilon>0\), there exists a constant \(C_{\varepsilon} > 1\) depending only
on \((\gamma, v_2,\varepsilon)\) such that, if a point \(\bm{\xi}\in\cl{\Omega}\)
satisfies \({\rm{dist}}(\bm{\xi}, \, \Gamma_{\rm{sonic}}^5 \cup \Gamma_{\rm{sonic}}^6) > \varepsilon\),
then \(C_{\varepsilon}^{-1}\leq g(\bm{\xi})\leq C_{\varepsilon}\).

In fact, for property (<ref>) above, it suffices
to choose $d\defeq\min\{\tfrac{d_{\rm sep}}{2}, \hat{\delta}\}$,
with constants $d_{\rm sep}$ and $\hat{\delta}$ given by (<ref>) and (<ref>)
Then, choosing \(\varepsilon \defeq \tfrac{d}{2}\) in property (<ref>),
we deduce that there exists a constant \({C}_{\flat}>1\) depending only on \((\gamma, v_2)\) such that,
for any \(\btheta\in\Theta\),

\begin{equation}\label{Eq:FuncgEquivDistb}
{C}_{\flat}^{-1} {\rm{dist}}^\flat(\bm{\xi}, \, \Gamma^5_{\rm{sonic}}\cup \Gamma^6_{\rm{sonic}} ) \leq
g(\bm{\xi}) \leq
{C}_{\flat} \, {\rm{dist}}^\flat(\bm{\xi}, \, \Gamma^5_{\rm{sonic}}\cup \Gamma^6_{\rm{sonic}} )
\qquad
\text{for all $\bm{\xi}\in\cl{\Omega}$}\,,
\end{equation}

\begin{equation} \label{eq:dist-flat}
{\rm{dist}}^\flat(\bm{\xi}, \, \Gamma^5_{\rm{sonic}}\cup \Gamma^6_{\rm{sonic}} ) \defeq
\min \big\{ \frac{d}{2}, \, {\rm{dist}}_{5}(\bm{\xi}, \,\Gamma_{\rm{sonic}}^5), \,{\rm{dist}}_{6}(\bm{\xi}, \, \Gamma_{\rm{sonic}}^6) \big\}\,.
\end{equation}

There exists a constant \(\mu>0\) such that, if \(\varphi\) is an admissible solution corresponding
to \(\btheta\in\Theta\) and \(\Omega\) is its pseudo-subsonic region, then the pseudo-Mach number

\begin{equation*}
M(\bm{\xi})\defeq \frac{|D\varphi(\bm{\xi})|}{c(|D\varphi(\bm{\xi})|,\varphi(\bm{\xi})) }
\end{equation*}

\begin{equation} \label{eq:MachNumberNearSonicBoundary}
M^2(\bm{\xi}) \leq 1 - \mu g(\bm{\xi}) \qquad \text{in $\cl{\Omega}$}\,,
\end{equation}

and there exists a constant \(C>1\) such that

\begin{equation}\label{Eq:Ellipticity4PsedoPF}
C^{-1} {\rm{dist}}^\flat(\bm{\xi}, \, \Gamma^5_{\rm{sonic}}\cup \Gamma^6_{\rm{sonic}}) |\bm{\kappa}|^2
\leq \sum_{i,j=1}^2 \mathcal{A}^i_{p_j}(D\varphi(\bm{\xi}),\varphi(\bm{\xi}))\kappa_i\kappa_j \leq C|\bm{\kappa}|^2
\end{equation}

for all \(\bm{\xi}\in\overline{\Omega}\) and \(\bm{\kappa}=(\kappa_1,\kappa_2)\in\mathbb{R}^2\),
where \(\mathcal{A}(\bm{p},z) = \rho(\abs{\bm{p}},z)\bm{p},\)
\({\rm{dist}}^\flat(\blank, \blank)\) is given by (<ref>), and
constants \(\mu\) and \(C\) depend only on \((\gamma, v_2)\).

Let \(\varphi\) be any admissible solution corresponding to \(\btheta\in\Theta\)
with pseudo-subsonic region \(\Omega\) and curved transonic shock \(\Gamma_{\rm{shock}}\).

From Lemma <ref>, there exists a constant \(R>1\)
depending only on $(\gamma, v_2)$
such that

\begin{equation*}
\Omega \subseteq B_{R}(\bm{0})\,,
\qquad
\| c(|D\varphi|,\varphi) \|_{C^0(\cl{\Omega})}
\| g \|_{C^2(\cl{\Omega})}
\leq R\,,
\end{equation*}

where \(g(\cdot)\) is given by (<ref>).
Since \(O_5, O_6 \in \Gamma_{\rm sym}\),
\(\partial_\eta g = 0\) on \(\Gamma_{\rm{sym}}\).
Then, by <cit.>, we can choose
constants $C_0>1$, $\delta\in(0,\frac{3}{4C_0})$, and \(\mu_1 \in (0,1)\)
depending only on $(\gamma,v_2)$ such that, for any $\mu \in(0,\mu_1]$,
either inequality \(M^2+\mu g \leq C_0 \delta <1\) holds in \(\Omega\)
or the maximum of \(M^2 + \mu g\) over \(\cl{\Omega}\) cannot
be attained in \(\Omega \cup \Gamma_{\rm{sym}}\).

In the first case, (<ref>) follows immediately.
Thus, we focus on the second case that the maximum of \(M^2 + \mu g\) must
be attained on \(\partial \Omega \setminus \Gamma_{\rm{sym}}\).

We claim that there exists a constant \(\mu_2 \in (0,\mu_1]\) depending only
on \((\gamma,v_2)\) such that, whenever \(\mu \in (0,\mu_2]\),
\((M^2 + \mu g) \leq 1\) holds in \(\Omega\), from which (<ref>) follows.

Let the maximum of \(M^2 + \mu g\) be attained
at \(P_{\rm{max}}\in \partial \Omega \setminus \Gamma_{\rm{sym}}\),
and let \((M^2 + \mu g)(P_{\rm{max}}) > 1\).
Then, using Proposition <ref>, we may proceed as in Steps 2–3
of the proof of <cit.>
to find \(\mu_2 \in (0,\mu_1]\) depending only on \((\gamma,v_2)\)
such that \(P_{\rm max} \notin \Gamma_{\rm shock}\) when
\(\mu \in (0,\mu_2]\).
Therefore, \(P_{\rm max} \in \Gamma^5_{\rm sonic} \cup \Gamma^6_{\rm sonic}\).
However, reducing \(\mu_2\) depending only on \((\gamma,v_2)\) if necessary, we have
\begin{equation*}
(M^2 + \mu g)(P_{\rm max}) = \sup_{\bm{\xi} \in \Gamma^5_{\rm sonic} \cup \Gamma^6_{\rm sonic}} (M^2 + \mu g)(\bm{\xi}) \leq 1\,,
\end{equation*}

which is a contradiction.
In the final inequality above,
we have used that \(g|_{\Gamma^j_{\rm sonic}} = 0\) for \(\theta_{j-4} \in [0,\theta^{\rm s})\),
whilst \(g|_{\Gamma^j_{\rm sonic}} \leq C_{\flat} (c_j - \abs{D\varphi_j(P_0^{j-4})})\)
for \(\theta_{j-4} \in [\theta^{\rm s},\theta^{\rm d})\), for \(j = 5,6\).

§.§ Interior Hölder estimates away from the sonic boundaries

For any set $U\subseteq\mathbb{R}^2$ and any constant $\varepsilon>0$, define
the $\varepsilon$-neighbourhood of $U$ as

\begin{equation}\label{Eq:Def-N-eps}
\mathcal{N}_{\varepsilon}(U)\defeq \{\bm{\xi}\in\mathbb{R}^2 \,:\, {\rm dist}(\bm{\xi},\, U)<\varepsilon\}\,.
\end{equation}

From Proposition <ref>,
there exists a constant $C_{\rm sh}>0$ depending only on $(\gamma, v_2)$ such that

\begin{equation*}
{\rm dist}(\Gamma_{\rm shock}, \, \partial B_1(O_2))\geq C_{\rm sh}^{-1}
\end{equation*}

for any admissible solution and parameters \(\btheta\in\Theta\).
It follows that

\begin{equation*}
|D\varphi_2(\bm{\xi})|^2\geq1+d_0 \qquad
\text{for any $\bm{\xi}\in\mathcal{N}_{\tfrac{1}{2 C_{\rm sh}}}(\Gamma_{\rm{shock}})$}
\end{equation*}

for some constant $d_0>0$ depending only on $(\gamma, v_2)$.
Subsequently, we claim that there exist constants \(d_0'\in(0,d_0)\) and \(\varepsilon \in (0,\frac{1}{2 C_{\rm sh}})\)
depending only on \((\gamma,v_2)\) such that, for any admissible solution \(\varphi\),

\begin{equation}\label{PositiveDifference}
|D\varphi_2(\bm{\xi})|-|D\varphi(\bm{\xi})|\geq d_0'
\qquad\,\text{for any $\bm{\xi}\in\overline{\Omega}\cap\mathcal{N}_{\varepsilon}(\Gamma_{\rm shock})$}\,.
\end{equation}

Indeed, in the case: $\gamma>1$, from Definition <ref>(<ref>)–(<ref>)
and the Bernoulli law:

\[
\frac{1}{2}|D\varphi|^2+\varphi+\frac{\rho^{\gamma-1}-1}{\gamma-1}=B=\frac{1}{2}|D\varphi_2|^2+\varphi_2\,,
\]

we have

\[\begin{aligned} \frac{\gamma+1}{2}\left(\rho^{\gamma-1}-1\right)
\geq\, & \left(\rho^{\gamma-1}-1\right)+\frac{\gamma-1}{2}\left(|D\varphi|^2-1\right)\\
=\, & (\gamma-1)(\varphi_2-\varphi)+\frac{\gamma-1}{2}\left(|D\varphi_2|^2-1\right)\geq\frac{\gamma-1}{2}d_0\,,
\end{aligned}\]

\begin{equation*}
|D\varphi_2|^2-|D\varphi|^2 = \frac{2\left(\rho^{\gamma-1}-1\right)}{\gamma-1}+(\varphi-\varphi_2)\geq\frac{2d_0}{\gamma+1}+(\varphi-\varphi_2)\,.
\end{equation*}

Since $\varphi$ is Lipschitz continuous in $\overline{\Omega}$ from Lemma <ref>,
and $\varphi=\varphi_2$ on $\Gamma_{\rm shock}$, we obtain

\begin{equation*}
|D\varphi_2(\bm{\xi})|^2-|D\varphi(\bm{\xi})|^2 \geq\frac{d_0}{\gamma+1}
\qquad\text{for any $\bm{\xi}\in\overline{\Omega}\cap\mathcal{N}_{\varepsilon}(\Gamma_{\rm shock})$}\,,
\end{equation*}

with some $\varepsilon\in(0,\frac{1}{2C_{\rm sh}})$ sufficiently small.
Then (<ref>) follows from the above inequality and the uniform boundedness property in Lemma <ref>.
In the case: $\gamma=1$, it is straightforward to obtain (<ref>)
by using Definition <ref>(<ref>):
for any admissible solution $\varphi$,
$|D\varphi|^2\leq c^2(|D\varphi|,\varphi)\equiv1$ in $\overline{\Omega}$, since
the sonic speed is a constant.

For \(\bm{\xi} = (\xi,\eta) \in \mathbb{R}^2 \setminus \{O_2\}\), define the polar coordinates $(r,\theta)$ centred at \(O_2=(u_2,v_2)\) by

\begin{align*}
r(\cos\theta,\sin\theta) \defeq (\xi,\eta) - O_2\,.
\end{align*}

It follows from (<ref>) that

\begin{equation}\label{PositiveDifferenceDeriv2r}
\partial_r(\varphi_2-\varphi)\leq-(|D\varphi_2|-|D\varphi|)\leq -d_0' \qquad
\text{in $\overline{\Omega}\cap\mathcal{N}_{\varepsilon}(\Gamma_{\rm shock})$}\,.
\end{equation}

Therefore, we may apply the implicit function theorem to
express $\Gamma_{\rm shock}$ under these polar coordinates as

\begin{equation*}
\Gamma_{\rm shock}=\{\bm{\xi}(r,\theta)\,:\, r=f_{O_2,\rm{sh}}(\theta), \, \theta_{P_2} < \theta < \theta_{P_3} \}\,,
\end{equation*}

where $(f_{O_2,\rm{sh}}(\theta_{P_i}),\theta_{P_i})$ are the polar coordinates of points $P_i$, for $i=2,3$.
By Lemma <ref> and (<ref>),
there exists a constant $C_1>0$ depending only on $(\gamma, v_2)$ such that

\begin{equation*}
\|f_{O_2,\rm{sh}}\|_{C^{0,1}\left([\theta_{P_2},\theta_{P_3}]\right)}\leq C_1\,.
\end{equation*}

From the above, with the help of Lemma <ref>,
we deduce the following properties; the proof is similar to that of <cit.>.

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
Let $\varphi$ be an admissible solution corresponding to parameters $\btheta\in \Theta$.

There exists a constant $\delta^{\prime}>0$ depending only on $(\gamma, v_2)$ such that

\begin{equation*}
\partial_{\bm{\nu}}\varphi_2-\delta^{\prime}>\partial_{\bm{\nu}}\varphi\geq\delta^{\prime}
\qquad \text{on $\overline{\Gamma_{\rm shock}}$}\,,
\end{equation*}

where $\bm{\nu}=\frac{D(\varphi_2-\varphi)}{|D(\varphi_2-\varphi)|}$
is the unit normal vector on $\Gamma_{\rm shock}$ pointing into the interior of $\Omega$.

For each $d>0$ and $m=2,3,\cdots,$ there exist positive constants $s$ and $C_m$
depending only on $(\gamma, v_2,d)$ such that,
whenever $P = \bm{\xi}(r_P,\theta_{P}) \in \Gamma_{\rm shock}$ satisfies
${\rm{dist}}(P,\Gamma_{\rm sonic}^{5}\cup\Gamma_{\rm sonic}^{6})\geq d,$

|f^(m)_O_2,sh(θ_P)| + sup_B_s(P)∩Ω |D^mφ|≤C_m .

From Lemmas <ref>, <ref>, and <ref>(<ref>),
we have the following a priori interior estimates.

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For each $d>0$ and $m=2,3,\cdots,$
there exists a constant $C_{m,d}>0$ depending only on $(\gamma, v_2,m,d)$ such that
any admissible solution $\varphi$ corresponding to $\btheta\in\Theta$ satisfies

φ_m,Ω∩{dist(ξ,Γ_sonic^5∪Γ_sonic^6)>d} ≤C_m,d .

§.§ Weighted Hölder estimates near the sonic boundaries

From Definition <ref>, the sonic boundary $\Gamma_{\rm sonic}^{5}$ depends continuously
on $\theta_1\in[0,\theta^{\rm d})$ and shrinks into a single point $P_0^1=P_1=P_2$
when $\theta_1\in[0,\theta^{\rm s})$ tends to $\theta^{\rm s}$ from below (if additionally $v_2 \in [v_2^{\rm s},0)$).
Moreover, from Proposition <ref>,
the ellipticity of equation (<ref>) degenerates on the sonic boundary
$\Gamma_{\rm sonic}^5$ for $\theta_1\in[0,\theta^{\rm s})$,
while equation (<ref>) is uniformly elliptic in the case that
$\theta_1\in(\theta^{\rm s},\theta^{\rm d})$ up to $\Gamma_{\rm sonic}^5$ away from $\Gamma_{\rm sonic}^6$.

The above reasons lead us to consider the following four cases separately
for the weighted Hölder estimates of the admissible solutions near $\Gamma^5_{\rm sonic}$.
Since $\theta^{\rm s}$ can be equal to $\theta^{\rm cr}$ depending on \((\gamma,v_2)\),
some of the following cases may not
occur (cf. Remark <ref>(<ref>)):

* Case 1.  ${\theta}_{1}\in[0,{\theta}^{\rm{s}})$ away from ${\theta}^{\rm{s}}$;

* Case 2.  ${\theta}_{1}\in[0,{\theta}^{\rm{s}})$ close to ${\theta}^{\rm{s}}$;

* Case 3.  ${\theta}_{1}\in[{\theta}^{\rm{s}},{\theta}^{\rm{d}})$ close to ${\theta}^{\rm{s}}$;

* Case 4.  ${\theta}_{1}\in[{\theta}^{\rm{s}},{\theta}^{\rm{d}})$  away from ${\theta}^{\rm{s}}$.

Corresponding weighted $C^{2,\alpha}$–estimates near \(\Gamma^{6}_{\rm sonic}\) for the four cases with respect
to \(\theta_2 \in [0,\theta^{\rm d})\) are obtained immediately due to the symmetry of our problem.

For any \(\btheta \in \Theta\), define an open set $\Omega^5 \subseteq \Omega$

\begin{equation*}
\Omega^5 \defeq
\big(\Omega\cap\{\bm{\xi}\in\mathbb{R}^2_+ \,:\, \xi > u_5 + q_5\sin\theta_{25} \}\big)\setminus \overline{B_{c_5^*}(O_5)}\,,
\end{equation*}

where $q_5={\rm dist}(O_5,S_{25})$ is given by (<ref>), and
$c_5^* \defeq \frac{1}{2} \big(|O_5P_2|+{\rm dist}(O_5,S_{25})\big)$.
For \(\bm{\xi}\in \cl{\Omega^5}\), we introduce the modified polar coordinates \((x,y)\) centred at \(O_5\) defined by

\begin{equation}\label{PolarCoordiNearO5}
(x,y) \defeq \rcal (\xi,\eta)
\quad \Longleftrightarrow
\quad
(\xi,\eta)-O_5 = (c_5-x)(\cos y,\sin y) \qquad \text{with $y \in [0,\tfrac\pi2]$}\,.
\end{equation}

It is clear that $\rcal(\Omega^5) \subseteq \{(x,y) \,:\, 0<x<c_5-c_5^*\}$.
For the pseudo-potential $\varphi_5$ given by (<ref>), we have

\begin{equation*}
\varphi_5 \circ \rcal^{-1} (x,y) = -\frac12 (c_5-x)^2 + \frac12 u_5^2 - u_5 \xi^{P_0^1}
\qquad\text{for $(x,y) \in
\rcal(\Omega^{5})$}\,.
\end{equation*}

Note that \(u_5\xi^{P_0^1} \to -v_2 \eta_0 \) as \(\theta_1 \to 0^+\), where \(\eta_0 >0 \)
is defined in <ref>.

For any admissible solution $\varphi$ corresponding to parameters $\btheta\in \Theta$, let $\psi$ be given by

\begin{equation*}
\psi(x,y) \defeq (\varphi-\varphi_5) \circ \rcal^{-1}(x,y) \qquad \text{for $(x,y) \in \rcal(\Omega^5)$}\,.
\end{equation*}

Then $\psi$ satisfies a free boundary problem for an elliptic equation:

* Equation for $\psi$ in $\rcal(\Omega^5)$:
Direct calculation gives

\begin{equation}\label{Eq4PsiInNewCoordi}
\mathcal{N}(\psi)=\big(2x-(\gamma+1)\psi_x+\co_1\big)\psi_{xx}+\co_2\psi_{xy}
\end{equation}

with $\co_j=\co_j(\psi_x, \psi_y,\psi,x,c_5)$ for $j=1,\cdots,5$, defined as follows:

\begin{equation}\label{Eq:Def-co-k}
\begin{aligned}
\co_1(p_x,p_y,z,x,c) &= -\frac{x^2}{c} + \frac{\gamma+1}{2c} (2x-p_x) p_x - \frac{\gamma-1}{c} \Big( z + \frac{p_y^2}{2(c-x)^2} \Big)\,,\\
\co_2(p_x,p_y,z,x,c) &= -\frac{2(p_x+c-x)p_y}{c(c-x)^2}\,,\\
\co_3(p_x,p_y,z,x,c) &= \frac{1}{c(c-x)^2} \Big( (2c-x)x
- (\gamma-1)\big( z+(c-x)p_x - \frac{p_x^2}{2}\big) + \frac{(\gamma-1)p_y^2}{2(c-x)^2} \Big)\,,\\
\co_4(p_x,p_y,z,x,c) &= \frac{1}{c-x} \Big( x-\frac{\gamma-1}{c} \big( z+(c-x)p_x + \frac{p_x^2}{2}
+ \frac{(\gamma+1)p_y^2}{2(\gamma-1)(c-x)^2} \big) \Big)\,,\\
\co_5(p_x,p_y,z,x,c) &= -\frac{2(p_x+c-x)p_y}{c(c-x)^3}\,.
\end{aligned}
\end{equation}

* Boundary condition for $\psi$ on $\rcal(\Gamma_{\rm shock} \cap \partial\Omega^5)$:
Denote the Rankine–Hugoniot condition for \(\varphi\) on $\Gamma_{\rm shock}$ as

\begin{equation}\label{Def4FreeBdryFunc}
g^{\rm sh}(D\varphi,\varphi,\bm{\xi})\defeq \rho(|D\varphi|,\varphi)D\varphi\cdot\bm{\nu}-D\varphi_{2}\cdot\bm{\nu}
\end{equation}

where $\mathcal{A}(\bm{p},z)=\rho(|\bm{p}|,z)\bm{p}$,
and $\bm{\nu}=\frac{D(\varphi_2-\varphi)}{|D(\varphi_2-\varphi)|}$
is the unit interior normal vector on $\Gamma_{\rm shock}$.
For constant $\delta^{\prime}>0$ given in Lemma <ref>, we define a
function $\zeta \in C^{\infty}(\mathbb{R})$ by

\begin{equation*}
\zeta(t) = \begin{cases}

t &\quad  \text{for} \; t \geq \tfrac{3\delta^{\prime}}{4}\,,\\
\tfrac{\delta^{\prime}}{2}
& \quad \text{for} \; t < \tfrac{\delta^{\prime}}{2}\,,
\end{cases}
\end{equation*}

satisfying $\zeta'(t)\geq0$ on $\mathbb{R}$. Also, we define an extension of
$g^{\rm sh}(\bm{p},z,\bm{\xi})$ onto $\mathbb{R}^2\times\mathbb{R} \times \overline{\Omega^{5}}$:

\begin{equation}\label{FBFunc:gshmod}
g_{\rm mod}^{\rm sh}(\bm{p},z,\bm{\xi})\defeq \big(\tilde{\mathcal{A}}(\bm{p},z)-D\varphi_2(\bm{\xi})\big)
\cdot\frac{D\varphi_2(\bm{\xi})-\bm{p}}{\zeta(|D\varphi_2(\bm{\xi})-\bm{p}|)}\,,
\end{equation}

where $\tilde{\mathcal{A}}(\bm{p},z)$ is given by Lemma <ref>, and
$g_{\rm mod}^{\rm sh}(\bm{p},z,\bm{\xi})=g^{\rm sh}(\bm{p},z,\bm{\xi})$ if $|D\varphi_2(\bm{\xi})-\bm{p}|\geq\frac{3\delta_1}{4}$.
Then we define

\begin{equation*}
M^{(\theta_1)}(\bm{p},z,\xi)\defeq g_{\rm mod}^{\rm sh}(\bm{p}+D\varphi_5,z+\varphi_5, \xi,
\xi \tan \theta_{25} + a_{25}
\end{equation*}

with $(D\varphi_5,\varphi_5)$ evaluated at $\bm{\xi}=(\xi,\,\xi \tan \theta_{25} + a_{25}
Note that, by Proposition <ref>(<ref>) and
Definition <ref>,
depend continuously on $\theta_1\in[0,\theta^{\rm d}]$
with the limit:
$\lim\limits_{\theta_1\to0^+}(\theta_{25}, a_{25})
=(\pi, \eta_0)$,
where $\eta_0>0$ is given by (<ref>).

We prescribe the free boundary condition for $\psi$ to be

\begin{equation*}
\mathcal{B}^{(\theta_1)}(\psi_x,\psi_y,\psi,x,y)=0 \qquad
\text{on $\rcal(\Gamma_{\rm shock} \cap \partial \Omega^5)$}\,,
\end{equation*}

where $\mathcal{B}^{(\theta_1)}$ is given by

\begin{equation}\label{DefBTheta1}
\mathcal{B}^{(\theta_1)}(p_x,p_y,z,x,y)\defeq \abs{D\bar{\phi}_{25}-(p_1,p_2)} M^{(\theta_1)}(p_1,p_2,z,\xi)\,,
\end{equation}

with function $\bar{\phi}_{25}(\bm{\xi}) \defeq (\varphi_2-\varphi_5)(\bm{\xi})
= v_2(\eta-\xi\tan\theta_{25}-a_{25})$,

\begin{equation*}
\xi = v_2\tan\theta_{25} + (c_5-x) \cos y\,, \qquad
\begin{pmatrix}p_1\\p_2\end{pmatrix}
=\begin{pmatrix}\cos y & \sin y \\
\sin y & -\cos y\end{pmatrix}
\begin{pmatrix}-p_x \\ -\frac{p_y}{c_5-x}\end{pmatrix}.
\end{equation*}

* Other conditions for $\psi$: From Problem <ref>(<ref>)–(<ref>)
and Definition <ref>(<ref>), $\psi$ satisfies

\begin{equation*}
\psi \geq 0 \,\,\,\, \text{in $\rcal(\Omega^5)$}\,,
\qquad \;\;
\psi = 0 \,\,\,\,  \text{on $\rcal(\Gamma_{\rm sonic}^{5})$}\,,
\qquad \;\;
\psi_y = 0 \,\,\,\, \text{on $\rcal(\Gamma_{\rm sym}\cap\partial\Omega^5)$}\,.
\end{equation*}

Fix \(\alpha \in (0,1)\).
Define the $\alpha$-th parabolic distance between points
$\bm{x}=(x,y), \; \tilde{\bm{x}}=(\tilde{x},\tilde{y})\in (0,\infty) \times \mathbb{R}$ to be

\begin{equation*}
\delta^{({\rm par})}_{\alpha}(\bm{x},\tilde{\bm{x}})\defeq \left(|x-\tilde{x}|^2+\max\{x,\tilde{x}\}|y-\tilde{y}|^2\right)^{\frac{\alpha}{2}}.
\end{equation*}

Then, given constants $\sigma>0$ and $m\in\mathbb{Z}_{+},$ the parabolic norms are defined as follows:

For any open set $\mathcal{D}\subseteq (0,\infty) \times \mathbb{R}$
and any function $u\in C^2(\mathcal{D})$ in the $(x,y)$–coordinates, define

\begin{equation*} \begin{aligned}
& \|u\|^{(\sigma),({\rm par})}_{m,0,\mathcal{D}}
\defeq \sum\limits_{0\leq k+l\leq m}\sup_{\bm{x}\in\mathcal{D}}
\Big(x^{k+\frac{l}{2}-\sigma}|\partial^{k}_{x}\partial^{l}_y u(\bm{x})|\Big)\,,\\
& [u]^{(\sigma),({\rm par})}_{m,\alpha,\mathcal{D}}
\defeq \sum\limits_{k+l=m}\sup_{\substack{\bm{x},\tilde{\bm{x}}\in\mathcal{D}\\
\bm{x}\neq\tilde{\bm{x}}}}\Big(\min\big\{x^{\alpha+k+\frac{l}{2}-\sigma},\tilde{x}^{\alpha+k+\frac{l}{2}-\sigma}\big\}
\frac{|\partial^{k}_{x}\partial^{l}_y u(\bm{x})-\partial^{k}_{x}\partial^{l}_y
u(\tilde{\bm{x}})|}{\delta^{({\rm par})}_{\alpha}(\bm{x},\tilde{\bm{x}})}\Big)\,,\\
&\|u\|^{(\sigma),({\rm par})}_{m,\alpha,\mathcal{D}}
\defeq\|u\|^{(\sigma),({\rm par})}_{m,0,\mathcal{D}}+[u]^{(\sigma),({\rm par})}_{m,\alpha,\mathcal{D}}\,.
\end{aligned}
\end{equation*}

For any \(a>0,\) fix an open interval $I\defeq (0,a)$. For a function $f\in C^2(I),$ define

\begin{equation*}
\begin{aligned}
&\|f\|^{(\sigma),({\rm par})}_{m,0,I}\defeq \sum\limits_{k=0}^{m}\sup_{x\in I} \Big(x^{k-\sigma}|\partial^{k}_{x}f(x)|\Big)\,,\\
&[f]^{(\sigma),({\rm par})}_{m,\alpha,I}\defeq \sup_{\substack{x,\tilde{x}\in I\\
x\neq\tilde{x}}}\Big(\min\big\{x^{\alpha+m-\sigma},\tilde{x}^{\alpha+m-\sigma}\big\} \frac{|\partial^{m}_{x}f(x)-\partial^{m}_{x}f(\tilde{x})|}{|x-\tilde{x}|^{\alpha}}\Big)\,,\\
&\|f\|^{(\sigma),({\rm par})}_{m,\alpha,I}\defeq \|f\|^{(\sigma),({\rm par})}_{m,0,I}+[f]^{(\sigma),({\rm par})}_{m,\alpha,I}\,.
\end{aligned} \end{equation*}

* In the case \(\sigma = 2,\) write $\|u\|^{({\rm par})}_{2,\alpha,\mathcal{D}} \defeq \|u\|^{(2),({\rm par})}_{2,\alpha,\mathcal{D}}$ and
$\|f\|^{({\rm par})}_{2,\alpha,I} \defeq \|f\|^{(2),({\rm par})}_{2,\alpha,I}$.

§.§.§ Case 1. Admissible solutions for ${\theta}_{1}<\theta^{\rm s}$ away from $\theta^{\rm s}$.

Fix $\gamma\geq1$ and $v_2\in(v_{\min},0)$.
For some constant \(\bar{\varepsilon} > 0\) depending only on \((\gamma,v_2),\) any \(\sigma \in (0,\theta^{\rm s}),\)
and any $\alpha \in(0,1),$ there exist both $\varepsilon \in(0, \bar{\varepsilon}]$ depending only on $(\gamma,v_2,\sigma)$
and $C>0$ depending only on $(\gamma,v_2,\alpha)$
such that any admissible solution $\varphi = \psi \circ \rcal + \varphi_5$ corresponding to
$\btheta\in \Theta \cap \{0 \leq \theta_1 \leq \theta^{\rm s}-\sigma\}$ satisfies

\begin{equation}\label{Result4theta1a}
\norm{\psi}_{2, \alpha, \rcal(\Omega^5_{\varepsilon})}^{(\mathrm{par})} +
\norm{f_{5, \mathrm{sh}}-f_{5,0}}_{2, \alpha,(0, \varepsilon)}^{(\rm par)} \leq C\,,
\end{equation}

where \(\Omega^{5}_{\varepsilon} \subseteq \Omega\) is a neighbourhood of \(\Gamma^5_{\rm sonic},\)
and functions ${f}_{5, \mathrm{sh}}$ and ${f}_{5,0}$ represent $\Gamma_{\rm shock}$ and $S_{25}$ respectively
to be defined in Step 1 below.

The proof of this proposition is similar to that of <cit.>.
We sketch the main steps of the proof, while omitting most

1. The depiction and the properties of the reflected shock $S_{25}$ and
the free boundary $\Gamma_{\rm shock}$ in a small neighbourhood of the sonic boundary $\Gamma^5_{\rm sonic}$.
It follows from the geometric properties of $\overline{\Omega^5}$ that
there exist positive constants $(\varepsilon_1,\varepsilon_0,\omega_0)$ depending only on $(\gamma,v_2)$,
with $\varepsilon_0<\varepsilon_1$ and \(\omega_0 \in (0,1)\),
and a unique function ${f}_{5,0}\in C^{\infty}([-\varepsilon_0,\varepsilon_0])$ satisfying

\begin{align*}\label{Eq:PreLemBSP5}
\rcal \big( S_{25}\cap\mathcal{N}_{\varepsilon_1}(\Gamma^5_{\rm sonic})
\big)
\cap\{|x|<\varepsilon_0\}
=\big\{(x,y) \,:\, |x|<\varepsilon_0, \, y={f}_{5,0}(x)\big\}\,,
\end{align*}

where $\mathcal{N}_{\varepsilon_1}(\Gamma^5_{\rm sonic})$ is defined by (<ref>),
and $2\omega_0\leq{f}'_{5,0}(x)\leq\omega_0^{-1}$ for $x\in(-\varepsilon_0,\varepsilon_0)$.
Then, for any $\varepsilon\in(0,\varepsilon_1]$, we define a set $\Omega^5_{\varepsilon} \subseteq \Omega$ by

\begin{equation}\label{Eq:Def4Omega5epsilon}
\Omega^5_{\varepsilon}\defeq \Omega\cap\mathcal{N}_{\varepsilon_1}(\Gamma^{5}_{\rm sonic})\cap \rcal^{-1} \big(
\{x<\varepsilon\}
\big).
\end{equation}

Using (<ref>) and (<ref>),
we can show that there exist $\bar{\varepsilon}\in(0,\varepsilon_0]$, $L\geq1$, $\delta\in(0,\frac12)$,
and $\omega\in(0,\omega_0)$ depending only on $(\gamma,v_2)$ such that,
for all $(x,y)\in \rcal(\Omega^5_{\bar{\varepsilon}})$,

\begin{equation}\label{Eq:PrepLemASP05}
0 \leq \psi(x,y) \leq Lx^2\,, \qquad 0\leq\psi_x\leq\frac{2-\delta}{1+\gamma}x\leq Lx\,,
\qquad
|\psi_y(x,y)|\leq Lx\,,
\end{equation}

and there exists a unique function ${f}_{5,{\rm sh}}\in C^1([0,\bar{\varepsilon}])$ such that

\begin{align*}
(x,y) \,:\, 0 < x < \bar{\varepsilon}, \, 0<y<{f}_{5,{\rm sh}}(x)
\big\}\,,\\
&\rcal(\Gamma_{\rm shock}\cap \partial\Omega^5_{\bar{\varepsilon}})
(x,y) \,:\, 0 < x < \bar{\varepsilon}, \, y={f}_{5,{\rm sh}}(x)
\big\}\,,
\end{align*}

and $\omega\leq{f}'_{5,{\rm sh}}(x)\leq L$ for any $x\in(0,\bar{\varepsilon})$.

2. Ellipticity for the equation and the free boundary condition.
Equation (<ref>) for $\psi$
in $\rcal(\Omega^5)$ can be written as

\begin{equation}\label{QuasilinearEq4Theta1}
\sum_{i,j=1}^{2}\hat{A}_{ij}^{(\theta_1)}(\psi_x,\psi_y,\psi,x)
\partial_{i} \partial_{j} \psi+\sum_{i,j=1}^{2}\hat{A}_{i}^{(\theta_1)} \partial_i \psi=0\,,
\end{equation}

with $(\partial_1,\partial_2) =(\partial_x,\partial_y)$
and $\hat{A}^{(\theta_1)}_{12}=\hat{A}^{(\theta_1)}_{21}$.
Then there exist $\varepsilon_5\in(0,\frac{\bar{\varepsilon}}{4}]$ and $\lambda_5>0$
depending only on $(\gamma,v_2)$ such that, for all $(x,y)\in \rcal( \overline{\Omega^5_{4\varepsilon_5}})$,

\begin{equation*}\label{PreLemBCP0}
\frac{\lambda_5}{2}|\bm{\kappa}|^2
\leq \sum_{i,j=1}^{2}\hat{A}^{(\theta_1)}_{ij}(\psi_x (x,y), \psi_y (x,y), \psi(x,y),x)\frac{\kappa_i\kappa_j}{x^{2-\frac{i+j}{2}}}
\leq\frac{2}{\lambda_5}|\kappa|^2 \qquad \text{for all $\bm{\kappa}\in\mathbb{R}^2$}\,.
\end{equation*}

Moreover, $\mathcal{B}^{(\theta_1)}$ defined by (<ref>)
satisfies $\mathcal{B}^{(\theta_1)}(\mathbf{0},0,x,y)=0$ for all $(x,y)\in\mathbb{R}^2$.
Let $y_{P_2}$ be the $y$–coordinate of point $P_2$ defined in Definition <ref>.
For each $m=2,3,\cdots$, there exist constants $\delta_{\rm bc}>0$ and $C>1$ depending only on $(\gamma,v_2,m)$
such that, whenever $|(p_x,p_y,z,x)|\leq\delta_{\rm bc}$ and $|y-y_{P_2}|\leq\delta_{\rm bc}$,

\begin{equation*}\label{PreLemBCP2}
\big|D^{m}\mathcal{B}^{(\theta_1)}(p_x,p_y,z,x,y)\big|\leq C\,.
\end{equation*}

Furthermore, there exist  constants $\hat{\delta}_{\rm bc}>0$, $C>1$,
and $\varepsilon'>0$ depending only on $(\gamma,v_2)$ such that,
whenever $|(p_x,p_y,z,x)|\leq\hat{\delta}_{\rm bc}$ and $|y-y_{P_2}|\leq\hat{\delta}_{\rm bc}$,

\begin{equation*}
\begin{aligned}
& \partial_{a}\mathcal{B}^{(\theta_1)}(p_x,p_y,z,x,y) \leq -C^{-1}
&&\quad \text{for $\partial_a = \partial_{p_x}, \partial_{p_y}$, or $\partial_z$}\,,\\
& D_{(p_x,p_y)}\mathcal{B}^{(\theta_1)}(p_x,p_y,z,x,y)\cdot\bm{\nu}^{(x,y)}_{\rm sh} \geq C^{-1}
&&\quad \text{on $\rcal(\Gamma_{\rm shock}\cap\partial\Omega_{\varepsilon'}^{5})$}\,,
\end{aligned}
\end{equation*}

the vector field $\bm{\nu}^{(x,y)}_{\rm sh}$ is the unit normal vector to $\rcal(\Gamma_{\rm shock})$
expressed in the $(x,y)$–coordinates and oriented towards the interior of $\rcal(\Omega)$.

3. Extension of the domain for the coefficients to equation (<ref>).
Let $\varepsilon_0>0$ and $L\geq1$ be the constants defined as above.
Then there exist constants $\varepsilon\in(0,\frac{\varepsilon_0}{2}]$
and $C>0$ depending only on $(\gamma,v_2)$ such that,
for any admissible solution $\varphi$
corresponding to parameters $\btheta\in \Theta$, function \(\psi = (\varphi - \varphi_5)\circ\rcal^{-1}\) satisfies

\begin{equation}\label{ModQuasilinearEq4Theta1}
\sum\limits_{i,j=1}^{2}\hat{A}_{ij}^{(\rm mod)}(\psi_x, \psi_y, \psi,x) \partial_{i} \partial_{j} \psi
+\sum\limits_{i,j=1}^2\hat{A}_{i}^{(\rm mod)}(\psi_x ,\psi_y, \psi,x) \partial_{i} \psi=0\qquad \text{in $\rcal(\Omega^5_{\varepsilon})$}\,,
\end{equation}

with $(\partial_1,\partial_2)\defeq (\partial_x,\partial_y)$ and
coefficients $(\hat{A}_{ij}^{(\rm mod)},\hat{A}_{i}^{(\rm mod)})$ satisfying

\begin{equation*}
(\hat{A}_{ij}^{(\rm mod)},\hat{A}_{i}^{(\rm mod)})=(\hat{A}_{ij}^{(\theta_1)},\hat{A}_{i}^{(\theta_1)})
\quad\,\, \text{in $\{(p_x,p_y,z,x) \,:\, |(p_x,p_y)|\leq Lx, \, |z|\leq Lx^2, \, 0 < x < \varepsilon \}$}\,.
\end{equation*}

Moreover, the modified coefficients satisfy

\begin{align*}
& \big|(\hat{A}_{11}^{(\rm mod)},\hat{A}_{12}^{(\rm mod)},\hat{A}_{2}^{(\rm mod)})(p_x,p_y,z,x)\big|
\leq Cx \qquad\text{in $\mathbb{R}^2\times\mathbb{R}\times(0,\varepsilon)$}\,,\\
& \big\|(\hat{A}_{22}^{(\rm mod)},\hat{A}_{1}^{(\rm mod)})\big\|_{0,\mathbb{R}^2\times\mathbb{R}\times(0,\varepsilon)}
+\big\|D_{(p_x,p_y,z,x)}(\hat{A}_{ij}^{(\rm mod)},\hat{A}_{i}^{(\rm mod)})\big\|_{0,\mathbb{R}^2\times\mathbb{R}\times(0,\varepsilon)}\leq C\,.
\end{align*}

4. Rescaling method in the local domain to obtain the weighted Hölder estimate.
Using the strictly decreasing property of $\eta^{P_2}$ with respect
to $\theta_1 \in [0,\theta^{\rm s}]$ which is shown in Lemma <ref>(<ref>),
we obtain a constant $l_{\rm so}>0$ depending only on $(\gamma,v_2,\sigma)$ such that,
for any admissible solution $\varphi$ corresponding to