# Existence and regularity of solutions of a supersonic-sonic patch arising in
axisymmetric relativistic transonic flow with general equation of state

Rahul Barthwal  T. Raja Sekhar  Department of Mathematics, Indian Institute of
Technology Kharagpur, Kharagpur, India

###### Abstract

In this article, we prove the existence and regularity of a smooth solution
for a supersonic-sonic patch arising in a modified Frankl problem in the study
of three-dimensional axisymmetric steady isentropic relativistic transonic
flows over a symmetric airfoil. We consider a general convex equation of state
which makes this problem complicated as well as interesting in the context of
the general theory for transonic flows. Such type of patches appear in many
transonic flows over an airfoil and flow near the nozzle throat. Here the main
difficulty is the coupling of nonhomogeneous terms due to axisymmetry and the
sonic degeneracy for the relativistic flow. However, using the well-received
characteristic decompositions of angle variables and a partial hodograph
transformation we prove the existence and regularity of solution in the
partial hodograph plane first. Further, by using an inverse transformation we
construct a smooth solution in the physical plane and discuss the uniform
regularity of solution up to the associated sonic curve. Finally, we also
discuss the uniform regularity of the sonic curve.

###### keywords:

Supersonic-sonic patch, Characteristic decomposition, Relativistic Euler
equations, Modified Frankl problem, Transonic flows

###### MSC:

[] 35A01; 35B45; 35L50; 35L65; 35M30

##  1 Introduction

The transonic flow problems are one of the most important problems in
mathematical fluid dynamics since transonic flow appears in various important
physical phenomena. In the context of transonic flow problems, the study of
supersonic bubbles is of utter importance. For a compressible flow passing the
duct, Courant and Friedrichs in their famous book [1] described that if the
Mach number of the flow is not much below one, then the flow becomes
supersonic somewhere on the surface of the duct due to the convexity of the
duct and is again purely subsonic throughout the exit section. Similar
situations arise naturally in many engineering and aerospace applications,
such as the flow over an airfoil or in a flow through an axisymmetric nozzle;
see Figure 1. We refer readers to the monographs of Bers [2], Kuz’min [3] and
Shapiro [4] for more details on transonic flows.

Figure 1: Transonic flow over an axisymmetric body or airfoil

In the last century, a large number of significant contributions have been
made in order to prove the existence of the global transonic solution to such
transonic flow problems, but it remains an open mathematical problem till now.
The main complexity of the transonic flow is that a transonic structure
consists of subsonic and supersonic parts, which are separated by either a
sonic curve or transonic shock. These are usually free boundaries due to the
nonlinearity of the governing system. Not only this but also the governing
systems of transonic flows can change their behavior across the sonic boundary
and are usually linearly degenerate on the sonic curve; see [5, 6, 7]. Such
features of transonic flow are more complicated to handle when compared to a
study of purely subsonic or supersonic flow.

A lot of important existence results for the subsonic-sonic part of the
transonic flow for steady Euler equations have been developed in the recent
years. Gilbarg and Serrin [8] provided a uniqueness result for a subsonic flow
past an axisymmetric body, while Xie and Xin [9, 10] proved the existence of
global subsonic-sonic solutions for a 3-D axially symmetric nozzle. In [11],
Chen et al. established the global existence of a subsonic-sonic solution for
the full Euler equations using the compensated-compactness framework.
Recently, Wang and Xin [12] proved the existence and uniqueness of a solution
for smooth transonic flows of Meyer type in de Laval nozzles and also obtained
the first result on the well-posedness for general subsonic-sonic flow
problems in [13]. On the other hand, for the supersonic part a sonic-
supersonic solution for steady isentropic Euler equations was constructed by
Zhang and Zheng [14] while Hu and Li proved the existence of a sonic-
supersonic solution for 2-D steady and pseudo-steady full Euler equations [15,
16]. The partial hodograph transformation used in the works of Hu and Li; viz.
[17, 15, 18] has become very crucial while solving sonic-supersonic boundary
value problems. We refer readers to [19, 20, 21, 22, 23, 17, 24] and
references cited therein for more such results in the context of sonic-
subsonic and sonic-supersonic boundary value problems.

Figure 2: Transonic flow in a channel with a supersonic bubble

Morawetz, in his work on transonic flow in a channel or a duct, (see Figure 2)
indicated that a smooth transonic flow does not exist in general, which means
that there may exist a transonic shock in the downstream flow [25]. However,
it is of utter importance to construct shock-free transonic flows. In [26],
Frankl explored the transonic flow over a symmetric airfoil and suggested that
a smooth transonic flow may exist if the part of the airfoil is free of
boundary conditions. The original Frankl problem is formulated to find the
smooth airfoil’s arc $\widehat{EG}$ when the slip conditions on the arcs
$\widehat{PE}$ and $\widehat{GQ}$ are prescribed; see Figure 3. Many existence
and uniqueness results for this Frankl problem have been discussed in the last
century; see viz. [27, 28]. Kuz’min [3] proposed a modified Frankl problem in
which a velocity distribution is prescribed on the arcs $\widehat{PE}$ and
$\widehat{GQ}$ instead of the slip boundary conditions. From a physical point
of view, such problems describe the transonic flows past permeable boundaries.
The modified Frankl problem can be utilized in many industrial applications as
well, where the design of the airfoil and wing usually needs to be formulated
according to some specific requirements of the aircraft; for example, one may
require the wing profile to have a particular velocity distribution, a
specific lift distribution or a certain temperature or pressure distribution.
Such a method in the area of aircraft design is usually known as the inverse
design method which was pioneered by Lighthill [29] and then developed further
by many researchers working in this field; see viz. [30, 31, 32, 33, 34, 35,
36] and references cited therein.

Figure 3: Modified Frankl problem for a transonic flow over a symmetric
airfoil: If the velocity distribution on the arcs $\widehat{PE}$ and
$\widehat{GQ}$ are prescribed, then find the airfoil’s arc $\widehat{EG}$ free
of boundary conditions for the correctness of the problem in the class of
smooth solutions

The modified Frankl problem has been studied extensively in the recent past.
Kuz’min discussed the existence and uniqueness of the solution of the modified
Frankl problem for a linearized version of the von Karman equation in a finite
domain; see [37, 38]. Recently, Hu and Li established the existence and
regularity of solutions of a sonic-supersonic patch extracted from a modified
Frankl problem for 2-D steady isentropic Euler equations and 3-D steady
axisymmetric isentropic Euler equations with ideal gas [39, 40]. The recent
development in the context of transonic flows has motivated us to ask
naturally whether such analysis can be performed for more complicated mixed-
type systems for a more general equation of state or not. Inspired by this
idea, the main motivation to do this work is to develop the existence and
regularity of a sonic-supersonic solution arising in a modified Frankl problem
for 3-D steady axisymmetric relativistic Euler equations with a general convex
equation of state.

In the domain of astrophysics, plasma physics, and nuclear physics, the
velocity of fluid particles are usually very large and often very close to the
speed of light as well, which means that the relativistic effects have to be
taken into consideration and the classical Euler equations of gas dynamics are
no longer valid. In the case of such high-speed flow, the governing system
under consideration is referred to as the relativistic Euler system. In the
recent years, a lot of interesting work has been done in the context of
relativistic gas dynamics systems; see viz. [41, 42, 43] and references cited
therein. Here, we consider the three-dimensional steady isentropic
relativistic Euler equations of the form

$\begin{cases}\begin{aligned} &(n\gamma u_{1})_{x}+(n\gamma
u_{2})_{y}+(n\gamma u_{3})_{z}=0,\\\
&((\rho+p)\gamma^{2}u_{1}^{2}+p)_{x}+((\rho+p)\gamma^{2}u_{1}u_{2})_{y}+((\rho+p)\gamma^{2}u_{1}u_{3})_{z}=0,\\\
&((\rho+p)\gamma^{2}u_{1}u_{2})_{x}+((\rho+p)\gamma^{2}u_{2}^{2}+p)_{y}+((\rho+p)\gamma^{2}u_{2}u_{3})_{z}=0,\\\
&((\rho+p)\gamma^{2}u_{1}u_{3})_{x}+((\rho+p)\gamma^{2}u_{2}u_{3})_{y}+((\rho+p)\gamma^{2}u_{3}^{2}+p)_{z}=0,\end{aligned}\end{cases}$
(1.1)

where $n$ denotes the proper number density, $u_{1},u_{2}$ and $u_{3}$ are the
velocity components of the velocity along $x,y$ and $z$ directions,
respectively.
$\gamma=\left(\sqrt{1-u_{1}^{2}-u_{2}^{2}-u_{3}^{2}}\right)^{-1}\leavevmode\nobreak\
(0<u_{1}^{2}+u_{2}^{2}+u_{3}^{2}<1)$ is the Lorentz factor such that the speed
of light is normalized to be 1, $\rho+p(\rho)=i$ denotes the enthalpy per unit
volume and $p=p(\rho)$ is the pressure of the relativistic fluid with $\rho$
being the total mass-energy density. In the cylindrical coordinates
$(x,r,\sigma)$ a flow is said to be axisymmetric if the state variables are
independent of the angle $\sigma$. Further, if we consider that the flow is
axisymmetric about the $x$ axis and is without swirl, i.e.,
$(\rho,u_{1},u_{2},u_{3})$ satisfy

$\displaystyle\begin{cases}\rho(x,y,z)=\rho(x,r),\leavevmode\nobreak\
\leavevmode\nobreak\ n(x,y,z)=n(x,r),\leavevmode\nobreak\ \leavevmode\nobreak\
u_{1}(x,y,z)=u(x,r),\\\ u_{2}(x,y,z)=v(x,r)\cos\sigma,\leavevmode\nobreak\
\leavevmode\nobreak\ u_{3}(x,y,z)=v(x,r)\sin\sigma,\end{cases}$

where $u$ and $v$ are the axial and radial velocity components, respectively.

The system (1.1) can be now rewritten in terms of $(\rho,u,v)(x,r)$ as
follows:

$\displaystyle\begin{cases}&(n\gamma u)_{x}+(n\gamma v)_{r}=-\dfrac{n\gamma
v}{r},\\\
&((\rho+p)\gamma^{2}u^{2}+p)_{x}+((\rho+p)\gamma^{2}uv)_{r}=-\dfrac{(\rho+p)\gamma^{2}uv}{r},\\\
&((\rho+p)\gamma^{2}uv)_{x}+((\rho+p)\gamma^{2}v^{2}+p)_{r}=-\dfrac{(\rho+p)\gamma^{2}v^{2}}{r},\end{cases}$
(1.2)

where $\gamma=(1-q^{2})^{-1/2}$ is the normalized Lorentz factor of
axisymmetric relativistic flow and $0<q=\sqrt{u^{2}+v^{2}}<1$ is the flow
velocity. Further, throughout the article we assume that the mass-energy
density $\rho$ and pressure $p=p(\rho)$ satisfies [43, 44]

$\displaystyle 0<\rho<\rho_{\max}<\infty,\leavevmode\nobreak\
\leavevmode\nobreak\ 0<p^{\prime}(\rho)<1,\leavevmode\nobreak\
\leavevmode\nobreak\ p^{\prime\prime}(\rho)>0.$ (1.3)

and $p^{\prime\prime}(\rho)$ remains finite for all values of $\rho$, which is
generally true for all physically relevant equations of states.

Now noting the Figure 3, we define the sonic-supersonic problem under
consideration for 3-D axisymmetric relativistic Euler equations precisely as
follows:

Supersonic-sonic boundary value problem extracted from modified Frankl problem
for 3-D relativistic flow:
If $\widehat{PE}$ is an increasing and concave smooth streamline of an
axisymmetric relativistic transonic flow and a velocity distribution is
prescribed on the arc $\widehat{PE}$ such that the point $P$ is sonic, then
find a sonic curve $\widehat{PD}$ starting from point $P$ and construct a
smooth supersonic solution for 3-D axisymmetric steady isentropic relativistic
Euler equations in a region $PFD$ near the point $P$ bounded by the sonic
curve $\widehat{PD}$, the arc $\widehat{PF}$ and a negative characteristic
curve $\widehat{DF}$ for a general convex pressure. Moreover, check the
regularity of the constructed solution.

One of the main complexity of the problem under consideration is that the
velocity data is given only on the streamline arc $\widehat{PE}$ in contrast
to all other sonic-supersonic boundary value problems or semi-hyperbolic patch
problems where the data is prescribed not only on a streamline but also on a
characteristic curve as well (see for example [18, 17, 45, 46]). In
particular, for relativistic Euler equations, we refer readers to [47]. The
other important complexity in this problem is to handle the nonhomogeneous
terms due to the axisymmetry and the sonic degeneracy along the sonic curve.
In all the previous work related to 2-D steady systems, angle variables (Mach
angle and flow angle) were chosen as independent variables to convert the
governing system into a linearized one. However, one can not expect to
linearize the axisymmetric systems due to the presence of nonhomogeneous
terms. To overcome these complexities, we use partial hodograph transformation
where the independent variables are Mach angle and the potential function to
convert the governing axisymmetric relativistic Euler system into a new
degenerate hyperbolic system. The idea of choosing such independent variables
is taken from a very recent work of Hu [40]. However, unlike the Mach-flow
angle plane, the reduced hyperbolic equations in our case do not form a closed
system and additional equations are needed to be added to the system in order
to close the system which makes the current problem even more complicated. We
also comment that the derivation of a priori estimates of solutions for the
current problem is also not very easy as a priori estimates developed in the
previous works such as [48, 46, 6, 7, 49] which are based on characteristic
decompositions in homogeneous form. But the nonhomogeneous terms in this
problem lead us to the nonhomogeneous form of characteristic decompositions of
the angle variables, which greatly affect the establishment of a priori
estimates of the solutions. However, using some proper auxiliary functions and
characteristic decompositions on them, we are able to develop the $C^{0}$ and
$C^{1}$ estimates of the solutions of this new degenerate hyperbolic system in
the partial hodograph plane, which helps us to develop a global solution and
its regularity in the partial hodograph plane. Finally, using an inverse
transformation, we transform these solutions back to the physical plane in
order to solve the original problem.

The rest of the article can be organized in the following manner. In section
2, we discuss the basic properties of the axisymmetric steady isentropic
relativistic Euler equations (1.2) and define the characteristic angles for
relativistic flow. Section 3 is devoted to defining the problem precisely and
prescribing the boundary data on the arc $\widehat{PE}$. Using a partial
hodograph transformation, we discuss the existence and regularity of solutions
to sonic-supersonic boundary value problem in the new coordinate system in
section 4. In section 5, we transform the constructed solutions back into the
physical plane by using an inverse transformation and verify that the
solutions constructed actually satisfy the boundary value problem. Finally, we
provide conclusions and the future scope of this work in section 6.

##  2 Basic properties of three-dimensional axisymmetric isentropic
irrotational steady relativistic Euler equations

We assume that the relativistic flow is irrotational, i.e., $u_{r}=v_{x}$ then
by first equation of (1.2) and making use of
$\gamma_{x}=\gamma^{3}qq_{x},\leavevmode\nobreak\ \leavevmode\nobreak\
\gamma_{r}=\gamma^{3}qq_{r},$ second equation of system (1.2) becomes

$\dfrac{n}{\gamma}\bigg{\\{}\dfrac{i}{n}\gamma_{x}+\dfrac{\gamma}{n}p_{x}+\dfrac{\gamma
u}{n}\bigg{[}n\gamma u\left(\dfrac{\gamma i}{n}\right)_{x}+n\gamma
v\left(\dfrac{\gamma i}{n}\right)_{r}\bigg{]}\bigg{\\}}=0.$ (2.1)

Further, by the second law of thermodynamics, we have

$\displaystyle d\left(\dfrac{i}{n}\right)=\dfrac{1}{n}dp+Tds,$ (2.2)

where $T$ is the absolute temperature and $s$ is the entropy of the flow.
Since the flow is assumed to be isentropic, i.e., $s$ is constant or in other
words $ds=0$. Therefore, in view of (2.1) and (2.2), we have

$\displaystyle\left(\dfrac{\gamma i}{n}\right)_{x}+\dfrac{\gamma
u}{n}\bigg{[}n\gamma u\left(\dfrac{\gamma i}{n}\right)_{x}+n\gamma
v\left(\dfrac{\gamma i}{n}\right)_{r}\bigg{]}=0.$ (2.3)

Similarly, from the third equation of system (1.2), one can easily obtain

$\displaystyle\left(\dfrac{\gamma i}{n}\right)_{r}+\dfrac{\gamma
v}{n}\bigg{[}n\gamma u\left(\dfrac{\gamma i}{n}\right)_{x}+n\gamma
v\left(\dfrac{\gamma i}{n}\right)_{r}\bigg{]}=0.$ (2.4)

It is easy to see that (2.3) and (2.4) form a homogeneous system of linear
equations for $\left(\dfrac{\gamma i}{n}\right)_{x}$ and $\left(\dfrac{\gamma
i}{n}\right)_{r}$. Now the determinant of the coefficient matrix is

$\displaystyle\begin{vmatrix}1+\gamma^{2}u^{2}&\gamma^{2}uv\\\
\gamma^{2}uv&1+\gamma^{2}v^{2}\end{vmatrix}=\dfrac{1}{1-q^{2}}\neq 0.$

Hence, we must have $\left(\dfrac{\gamma i}{n}\right)_{x}=\left(\dfrac{\gamma
i}{n}\right)_{r}=0$, which provides the Bernoulli’s law for axisymmetric
steady relativistic Euler equations of the form

$\displaystyle\dfrac{\gamma i}{n}=\mathrm{const.}$ (2.5)

For the convenience of the subsequent discussion, we write Bernoulli’s law in
the following form

$\displaystyle\dfrac{\gamma i}{n}=m\hat{\gamma},$ (2.6)

where $m$ is the average rest mass per particle and
$\hat{\gamma}^{-1}=\sqrt{1-\hat{q}^{2}}$ is a constant.

######  Lemma 2.1.

If $p$ satisfies $\dfrac{\partial p}{\partial n}>0$ for $n>0$ then there
exists a constant $\hat{q}\leavevmode\nobreak\ (0<\hat{q}<1)$ such that the
flow speed $q<\hat{q}$. The quantity $\hat{q}$ is called the limit speed of
the flow and the flow speed approaches the limit speed when $n$ approaches $0$
[43].

###### Proof.

Using the second law of thermodynamics for isentropic flow, we have

$\dfrac{d\left(\frac{i}{n}\right)}{dn}=\dfrac{d\left(\frac{i}{n}\right)}{dp}.\dfrac{dp}{dn}=\dfrac{1}{n}.\dfrac{dp}{dn}>0$

for $n>0$.

Therefore, using the Bernoulli’s law $\eqref{eq: 2.8}$ and the fact that
$\dfrac{i}{n}=\dfrac{mn+e+p}{n}\geq m$ for $n\geq 0$, it is easy to see that
$q<\hat{q}<1$ and $q$ approaches $\hat{q}$ as $n$ approaches $0$. ∎

Now noting the Bernoulli’s law (2.6), system (1.2) can be rewritten as

$\displaystyle\begin{cases}\gamma[(i\gamma u)_{x}+(i\gamma
v)_{r}]+i\gamma[u\gamma_{x}+v\gamma_{r}]=-\dfrac{i\gamma^{2}v}{r},\\\ \gamma
u[(i\gamma u)_{x}+(i\gamma v)_{r}]+i\gamma[u(\gamma u)_{x}+v(\gamma
u)_{r}]+p_{x}=-\dfrac{i\gamma^{2}uv}{r},\\\ \gamma v[(i\gamma u)_{x}+(i\gamma
v)_{r}]+i\gamma[u(\gamma v)_{x}+v(\gamma
v)_{r}]+p_{r}=-\dfrac{i\gamma^{2}v^{2}}{r},\end{cases}$ (2.7)

Then by taking the scalar product of (2.7) with $\gamma(1,-u,-v)$ and
simplifying, we obtain

$\displaystyle i(\gamma u)_{x}+i(\gamma v)_{r}=-\gamma u\rho_{x}-\gamma
v\rho_{r}-\dfrac{i\gamma v}{r}.$ (2.8)

Again from the momentum equations of (1.2), we can easily obtain

$\displaystyle\begin{cases}i\gamma^{2}uu_{x}+i\gamma^{2}vu_{r}+p_{x}=0,\\\
i\gamma^{2}uv_{x}+i\gamma^{2}vv_{r}+p_{r}=0.\end{cases}$ (2.9)

Then taking the scalar product of (2.9) with $\gamma(u,v)$, we have

$\displaystyle
i\gamma^{3}[u^{2}u_{x}+uv(u_{r}+v_{x})+v^{2}v_{r}]=-a^{2}(\gamma
u\rho_{x}+\gamma v\rho_{r}),$ (2.10)

where $a=\sqrt{p^{\prime}(\rho)}$ denotes the speed of sound relative to the
fluid.

Then by combining (2.8) and (2.10) the three-dimensional axisymmetric steady
isentropic irrotational relativistic flow can be governed by Bernoulli’s law
(2.6) and

$\displaystyle\begin{cases}(M_{1}^{2}-1)u_{x}+M_{1}M_{2}(u_{r}+v_{x})+(M_{2}^{2}-1)v_{r}=\dfrac{v}{r},\\\
u_{r}-v_{x}=0,\end{cases}$ (2.11)

where $M_{1}=\dfrac{\gamma u}{a\gamma_{a}},\leavevmode\nobreak\
\leavevmode\nobreak\ M_{2}=\dfrac{\gamma v}{a\gamma_{a}},$ and
$\gamma_{a}=\dfrac{1}{\sqrt{1-a^{2}}}$.

In matrix form (2.11) can be rewritten as

$\displaystyle\begin{bmatrix}M_{1}^{2}-1&M_{1}M_{2}\\\
0&-1\end{bmatrix}\begin{bmatrix}u\\\
v\end{bmatrix}_{x}+\begin{bmatrix}M_{1}M_{2}&M_{2}^{2}-1\\\
1&0\end{bmatrix}\begin{bmatrix}u\\\
v\end{bmatrix}_{r}=\begin{bmatrix}\dfrac{v}{r}\\\ 0\end{bmatrix}.$ (2.12)

Figure 4: $C_{\pm}$ characteristic directions, proper Mach angle and flow
angle

It is easy to see that the system (2.12) has the eigenvalues
$\Lambda_{\pm}=\dfrac{M_{1}M_{2}\pm\sqrt{M_{1}^{2}+M_{2}^{2}-1}}{M_{1}^{2}-1}$
with corresponding left eigenvectors $l_{\pm}=(1,\mp\sqrt{M^{2}-1})$, where
$M=\sqrt{M_{1}^{2}+M_{2}^{2}}=\dfrac{\gamma q}{\gamma_{a}a}$ is the proper
Mach number of the relativistic flow. The expression of these eigenvalues
shows that the system (2.12) is a mixed-type system and can change its
behavior from hyperbolic to elliptic across the sonic boundary ($M=1$);
therefore, its behavior depends on the choice of proper Mach number. For $M>1$
(supersonic) system (2.11) is hyperbolic while for $M<1$ (subsonic) it is
elliptic. Then we define the two families of wave characteristics as

$\dfrac{dr}{dx}=\Lambda_{\pm}.$ (2.13)

Moreover, we obtain the characteristic equations by multiplying $l_{\pm}$ to
the system $\eqref{eq: 2.11}$ as

$\displaystyle\begin{cases}{\partial}_{+}u+\Lambda_{-}{\partial}_{+}v=\dfrac{v}{r(M_{1}^{2}-1)},\\\
{\partial}_{-}u+\Lambda_{+}{\partial}_{-}v=\dfrac{v}{r(M_{1}^{2}-1)}\end{cases}$
(2.14)

where ${\partial}_{\pm}=\partial_{x}+\Lambda_{\pm}\partial_{r}$.

From the expression of eigenvalues $\Lambda_{\pm}$, it is easy to see that

$\displaystyle 1=\dfrac{|(M_{1},M_{2}).(\Lambda,-1)|}{|\Lambda,-1|},$ (2.15)

which means that the component of the flow velocity normal to $C_{\pm}$
characteristic curve is equal to $\dfrac{a\gamma_{a}}{\gamma}$. Then we define
the concept of characteristic direction as in [50]. The direction of the
characteristic is defined as the tangential direction that forms an acute
angle $\omega$ with the flow velocity vector $(u,v)$. Geometrically, the
$C_{+}$ characteristic direction forms the angle $\omega$ with the flow
velocity vector $(u,v)$ in a clockwise direction, while the $C_{-}$
characteristic direction forms the angle $\omega$ with the flow velocity
vector $(u,v)$ in the counterclockwise direction where $\omega$ is called the
proper Mach angle. Further, we denote the flow angle by $\theta$, which is the
angle between velocity vector $(u,v)$ and $x$-axis such that
$\tan\theta=\dfrac{v}{u}$ and $\sin\omega=\dfrac{1}{M}$ ; see Figure 4.

###  2.1 Characteristic equations in terms of characteristic angles

First we differentiate Bernoulli’s law (2.5) w.r.t. $q$, which yields

$\displaystyle\dfrac{\gamma^{3}qi}{n}+\gamma\dfrac{d(i/n)}{dp}\dfrac{dp}{da}\dfrac{da}{dq}=0.$
(2.16)

Then from $a^{2}=p^{\prime}(\rho)$ we have
$\dfrac{dp}{da}=\dfrac{2a^{3}}{p^{\prime\prime}(\rho)}$, which is exploited in
(2.16) along with second law of thermodynamics (2.2) to yield

$\displaystyle\dfrac{da}{dq}=-\dfrac{iq\gamma^{2}p^{\prime\prime}(\rho)}{2a^{3}}<0.$
(2.17)

Also, by $M=\dfrac{\gamma q}{a\gamma_{a}}$, it is easy to see that

$\displaystyle\dfrac{dM}{dq}=M\left(\dfrac{1}{q(1-q^{2})}-\dfrac{1}{a(1-a^{2})}\dfrac{da}{dq}\right).$
(2.18)

Then noting that $\dfrac{da}{dq}<0$ and $0<a^{2},q^{2}<1$ we have
$\dfrac{dM}{dq}>0$.

We invoke polar coordinates in velocity plane such that $u=q\cos\theta$ and
$v=q\sin\theta$. Then we can use the following formulas of velocity [43]

$u=\dfrac{a\gamma_{a}\cos\theta}{\gamma\sin\omega},\leavevmode\nobreak\
\leavevmode\nobreak\ v=\dfrac{a\gamma_{a}\sin\theta}{\gamma\sin\omega}.$
(2.19)

We further introduce the weighted directional derivatives along the
characteristics [51, 40]

$\displaystyle\begin{cases}&\tilde{\partial}_{+}=r\cos\alpha\partial_{x}+r\sin\alpha\partial_{r},\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{\partial}_{-}=r\cos\beta\partial_{x}+r\sin\beta\partial_{r},\\\
&\tilde{\partial}_{0}=r\cos\theta\partial_{x}+r\sin\theta\partial_{r},\end{cases}$
(2.20)

from which one has

$\displaystyle\partial_{x}=-\dfrac{\sin\beta\tilde{\partial}_{+}-\sin\alpha\tilde{\partial}_{-}}{r\sin(2\omega)},\leavevmode\nobreak\
\leavevmode\nobreak\
\partial_{r}=\dfrac{\cos\beta\tilde{\partial}_{+}-\cos\alpha\tilde{\partial}_{-}}{r\sin(2\omega)},\leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{\partial}_{0}=\dfrac{\tilde{\partial}_{+}+\tilde{\partial}_{-}}{2\cos\omega}.$
(2.21)

Then the characteristic equations (2.14) can be rewritten as

$\displaystyle\begin{cases}\tilde{\partial}_{+}u+\Lambda_{-}\tilde{\partial}_{+}v=\dfrac{v\cos\alpha}{M_{1}^{2}-1},\\\
\tilde{\partial}_{-}u+\Lambda_{+}\tilde{\partial}_{-}v=\dfrac{v\cos\beta}{M_{1}^{2}-1},\end{cases}$
(2.22)

Also, in terms of weighted directional derivatives, we have the first-order
decompositions of velocity components as

$\displaystyle\begin{cases}\tilde{\partial}_{\pm}u=\dfrac{\gamma_{a}}{\gamma\sin^{2}\omega}\bigg{[}\sin\omega\cos\theta
f(a)\tilde{\partial}_{\pm}a-a\sin\omega\sin\theta\tilde{\partial}_{\pm}\theta-a\cos\omega\cos\theta\tilde{\partial}_{\pm}\omega\bigg{]},\vspace{0.2
cm}\\\
\tilde{\partial}_{\pm}v=\dfrac{\gamma_{a}}{\gamma\sin^{2}\omega}\bigg{[}\sin\omega\sin\theta
f(a)\tilde{\partial}_{\pm}a+a\sin\omega\cos\theta\tilde{\partial}_{\pm}\theta-a\cos\omega\sin\theta\tilde{\partial}_{\pm}\omega\bigg{]},\end{cases}$
(2.23)

where
$0<f(a)=\left(\gamma_{a}^{2}+\dfrac{2a^{4}}{ip^{\prime\prime}(\rho)}\right)<\infty$.

Now using (2.23) in characteristic equation (2.22), we have

$\displaystyle\begin{cases}\tilde{\partial}_{+}\omega&=\dfrac{aF_{1}(a,\omega)}{2iq\gamma
p^{\prime\prime}(\rho)\cos\omega\gamma_{a}}\dfrac{\tilde{\partial}_{+}a}{a},\\\
\tilde{\partial}_{-}\omega&=\dfrac{aF_{1}(a,\omega)}{2iq\gamma
p^{\prime\prime}(\rho)\cos\omega\gamma_{a}}\dfrac{\tilde{\partial}_{-}a}{a},\\\
\tilde{\partial}_{+}\theta&=\dfrac{(F_{2}(a,\omega)-aF_{1}(a,\omega)\cos^{2}\omega)}{2iq\gamma
p^{\prime\prime}(\rho)\cos\omega\sin^{2}\omega\gamma_{a}}\dfrac{\tilde{\partial}_{+}a}{a}-\sin\omega\sin\theta,\\\
\tilde{\partial}_{-}\theta&=\dfrac{(aF_{1}(a,\omega)\cos^{2}\omega-
F_{2}(a,\omega))}{2iq\gamma
p^{\prime\prime}(\rho)\cos\omega\sin^{2}\omega\gamma_{a}}\dfrac{\tilde{\partial}_{-}a}{a}+\sin\omega\sin\theta,\end{cases}$
(2.24)

where

$\displaystyle\begin{cases}F_{1}(a,\omega)=2ip^{\prime\prime}(\rho)\gamma_{a}^{2}f(a)+4a^{2}\sin^{2}\omega>0,\\\
F_{2}(a,\omega)=iq\gamma\gamma_{a}p^{\prime\prime}(\rho)f(a)\cos\omega\sin
2\omega.\end{cases}$ (2.25)

If $\omega=\omega(a)$ then one can use
$\dfrac{d\omega}{da}=\dfrac{d\omega}{dM}\dfrac{dM}{dq}\dfrac{dq}{da}$ and
$\dfrac{da}{dq}<0,\leavevmode\nobreak\ \dfrac{dM}{dq}>0$ together with
$\sin\omega=\dfrac{1}{M}$ to yield $\dfrac{d\omega}{da}>0$. Let
$\varpi=\sin\omega,\omega\in[k_{0},\pi/2],\leavevmode\nobreak\ k_{0}>0$ is a
constant. Then by inverse function theorem we must have
$a=a(\omega)=a(\sin^{-1}\varpi)$. Therefore we can define
$F_{1}(a,\omega)=:F_{1}(\varpi)$ and $F_{2}(a,\omega)=:F_{2}(\varpi)$ to
obtain

$\displaystyle\begin{cases}\tilde{\partial}_{+}\theta+\dfrac{4a^{2}\cos\omega}{F_{1}(\varpi)}\tilde{\partial}_{+}\varpi=-\varpi\sin\theta,\\\
\tilde{\partial}_{-}\theta-\dfrac{4a^{2}\cos\omega}{F_{1}(\varpi)}\tilde{\partial}_{-}\varpi=\varpi\sin\theta.\end{cases}$
(2.26)

Due to the continuity of the function $\dfrac{2a^{2}(y)}{yF_{1}(y)},y\in[\sin
k_{0},1]$, we set

$\displaystyle I(\varpi)=\int_{\sin
k_{0}}^{\varpi}\dfrac{2a^{2}(y)}{yF_{1}(y)}dy,\leavevmode\nobreak\
\leavevmode\nobreak\ \varpi\in[\sin k_{0},1].$

Therefore, we write $I:=I(\varpi)$ to convert (2.26) in the following form

$\displaystyle\begin{cases}\tilde{\partial}_{+}\theta+\sin
2\omega\tilde{\partial}_{+}I=-\varpi\sin\theta,\\\
\tilde{\partial}_{-}\theta-\sin
2\omega\tilde{\partial}_{-}I=\varpi\sin\theta\end{cases}$ (2.27)

with

$\displaystyle\tilde{\partial}_{i}\varpi=\dfrac{\varpi
F_{1}(\varpi)}{2a^{2}(\varpi)}\tilde{\partial}_{i}I,\leavevmode\nobreak\
\leavevmode\nobreak\ i=0,\pm.$ (2.28)

Now we use the following commutator relation from [40]

$\displaystyle\tilde{\partial}_{-}\tilde{\partial}_{+}-\tilde{\partial}_{+}\tilde{\partial}_{-}$
$\displaystyle=\dfrac{1}{\sin 2\omega}\bigg{[}(\cos
2\omega\tilde{\partial}_{+}\beta-\tilde{\partial}_{-}\alpha)\tilde{\partial}_{-}-(\tilde{\partial}_{+}\beta-\cos
2\omega\tilde{\partial}_{-}\alpha)\tilde{\partial}_{+}\bigg{]}+\sin\beta\tilde{\partial}^{+}-\sin\alpha\tilde{\partial}^{-}$

to obtain the commutator relation of $I$ of the form

$\displaystyle\tilde{\partial}_{-}\tilde{\partial}_{+}I-\tilde{\partial}_{+}\tilde{\partial}_{-}I$
$\displaystyle=\dfrac{1}{\sin 2\omega}\bigg{[}(\cos
2\omega\tilde{\partial}_{+}\beta-\tilde{\partial}_{-}\alpha)\tilde{\partial}_{-}I-(\tilde{\partial}_{+}\beta-\cos
2\omega\tilde{\partial}_{-}\alpha)\tilde{\partial}_{+}I\bigg{]}+\sin\beta\tilde{\partial}^{+}I-\sin\alpha\tilde{\partial}^{-}I,$
$\displaystyle\tilde{\partial}_{-}\tilde{\partial}_{+}\theta-\tilde{\partial}_{+}\tilde{\partial}_{-}\theta$
$\displaystyle=\dfrac{1}{\sin 2\omega}\bigg{[}(\cos
2\omega\tilde{\partial}_{+}\beta-\tilde{\partial}_{-}\alpha)\tilde{\partial}_{-}\theta-(\tilde{\partial}_{+}\beta-\cos
2\omega\tilde{\partial}_{-}\alpha)\tilde{\partial}_{+}\theta\bigg{]}+\sin\beta\tilde{\partial}^{+}I-\sin\alpha\tilde{\partial}^{-}I.$

Therefore, if we denote $W=\tilde{\partial}_{+}I$, $Z=-\tilde{\partial}_{-}I$,
$f(a):=f(a(\varpi))$ and use
$F_{1}(\varpi)=4a^{2}\varpi^{2}+2ip^{\prime\prime}(\rho)\gamma_{a(\varpi)}^{2}f(a(\varpi))$,
then it is easy to obtain the characteristic decompositions of $W$ and $Z$ of
the form

$\displaystyle\begin{cases}&\tilde{\partial}_{-}W=W\Bigg{[}\dfrac{F_{1}(\varpi)}{4a^{2}\cos^{2}\varpi}(W-Z)+W-\cos
2\omega\leavevmode\nobreak\
Z+\dfrac{F_{1}(\varpi)}{2a^{2}}Z+2\sin\theta\cos\omega\Bigg{]}+\dfrac{\sin\beta}{2}(W-Z)+\sin^{2}\theta,\vspace{0.2
cm}\\\
&\tilde{\partial}_{+}Z=Z\Bigg{[}\dfrac{F_{1}(\varpi)}{4a^{2}\cos^{2}\varpi}(W-Z)-Z+\cos
2\omega\leavevmode\nobreak\
W-\dfrac{F_{1}(\varpi)}{2a^{2}}W+2\sin\theta\cos\omega\Bigg{]}-\dfrac{\sin\alpha}{2}(W-Z)-\sin^{2}\theta,\end{cases}$
(2.29)

which shows that the above equations form a nonhomogeneous system of equations
for $W$ and $Z$.

##  3 Formulation of the main problem and boundary data

We now formulate the problem mathematically in detail by mimicking the real
setting of the airfoil problem. Let
$\widehat{PE}:r=\varphi(x),x\in[x_{1},x_{2}]$, be a smooth curve and
$(\hat{u}(x),\hat{v}(x)),x\in[x_{1},x_{2}]$, is a given velocity distribution
on $\widehat{PE}$. Then we define our problem as follows

###  3.1 Main Problem

Let $\widehat{PE}:r=\varphi(x)\leavevmode\nobreak\ (x\in[x_{1},x_{2}])$ be a
smooth streamline of the three-dimensional axisymmetric steady relativistic
flow such that it is locally increasing and concave near the point $P$ and
$(\hat{u}(x),\hat{v}(x))$ is a given velocity distribution on $\widehat{PE}$
such that $M(x)>1\leavevmode\nobreak\ \forall x\in(x_{1},x_{3}]$ for some
$x_{3}\in(x_{1},x_{2}]$ and $M(x_{1})=1$. Then find a smooth sonic curve
$\widehat{PD}$ and build a smooth supersonic solution to system (2.11) in the
angular region of $P$ bounded by $\widehat{PE}$ and $\widehat{PD}$ ; see
Figure 3.

###  3.2 Reformulated Problem in terms of angle variables

We can actually reformulate our main problem in terms of angle variables
$(\theta,\varpi)$ as follows. From Bernoulli’s law (2.5) and the fact that
$\dfrac{da}{dq}<0$, it is easy to see that
$a=a(q(x))=a(\hat{u}^{2}(x)+\hat{v}^{2}(x))$. Then from
$\tan\theta=\dfrac{v}{u}$ and $\sin\omega=\dfrac{1}{M}$, we obtain the data
for angle variables $(\theta,\varpi)$ on $\widehat{PE}$ as

$\displaystyle\theta(x,\varphi(x))$
$\displaystyle=\tan^{-1}\left(\dfrac{\hat{v}(x)}{\hat{u}(x)}\right)=:\hat{\theta}(x),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \varpi(x,\varphi(x))$
$\displaystyle=\dfrac{\hat{a}(\hat{u}^{2}(x)+\hat{v}^{2}(x))\sqrt{1-\hat{a}^{2}(\hat{u}^{2}(x)+\hat{v}^{2}(x))}}{(\hat{u}^{2}(x)+\hat{v}^{2}(x))\sqrt{1-(\hat{u}^{2}(x)+\hat{v}^{2}(x))}}=:\hat{\varpi}(x).$
(3.1)

Then we reformulate our problem in terms of angle variables $(\theta,\varpi)$
as: Let us consider a locally increasing smooth streamline
$\widehat{PE}:r=\varphi(x)(x\in[x_{1},x_{2}])$ of three-dimensional
axisymmetric steady relativistic flow satisfying $\varphi^{\prime\prime}(x)<0$
in a neighbourhood of $x=x_{1}$ along which the angle variable $\varpi$
decreases and assign the boundary data
$(\theta,\varpi)=(\hat{\theta},\hat{\varpi})(x)$ on $\widehat{PE}$ such that
$\hat{\theta}(x)=\tan^{-1}\varphi^{\prime}(x),\leavevmode\nobreak\
\leavevmode\nobreak\ \hat{\varpi}(x)\in(0,1)\leavevmode\nobreak\ \forall
x\in(x_{1},x_{2}]$ and $\hat{\varpi}(x_{1})=1$. Then find a smooth sonic curve
$\widehat{PD}$ and build a smooth supersonic solution to system (2.26) in the
angular region of $P$ bounded by $\widehat{PE}$ and $\widehat{PD}$; see Figure
3.

In order to solve this problem, we assume that the functions $\varphi(x)$ and
$\hat{\varpi}(x)$ satisfy [40]

$\displaystyle\begin{cases}\varphi(x)\in C^{3}[x_{1},x_{2}],\hat{\varpi}(x)\in
C^{2}[x_{1},x_{2}],\\\ \varphi(x_{1})>0,\leavevmode\nobreak\
\leavevmode\nobreak\ \varphi^{\prime}(x_{1})>0,\leavevmode\nobreak\
\leavevmode\nobreak\ \varphi^{\prime\prime}(x_{1})<0,\leavevmode\nobreak\
\leavevmode\nobreak\ \hat{\varpi}^{\prime}(x_{1})<0,\end{cases}$ (3.2)

which implies that the curve $r=\varphi(x)$ is increasing and concave while
the angle variable $\varpi$ corresponding to the Mach number decreases near
the point $P$. One may note that these assumptions are consistent with the
real airfoil setting as well. Since we are focused to develop the existence
and regularity of solutions near point o$P$ only, therefore, one may assume
without loss of generality

$\displaystyle\begin{cases}\varphi(x)\in C^{3}[x_{1},x_{2}],\hat{\varpi}(x)\in
C^{2}[x_{1},x_{2}],\\\ \varphi_{0}\leq\varphi(x),\leavevmode\nobreak\
\leavevmode\nobreak\
\varphi^{\prime}(x)\leq\varphi_{1},\varphi^{\prime\prime}(x)<0,\hat{\varpi}^{\prime}(x)<0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ x\in[x_{1},x_{3}]\end{cases}$ (3.3)

for some $x_{3}\in(x_{1},x_{2}]$, where $\varphi_{0}$ and $\varphi_{1}$ are
some positive constants. We further assume that $\varphi(x)$ and
$\hat{\varpi}(x)$ satisfy

$\displaystyle\left(\dfrac{\varphi^{\prime\prime}}{1+(\varphi^{\prime})^{2}}-\dfrac{4a^{2}(\varpi)\sqrt{1-(\varpi)^{2}}}{F_{1}(\varpi)}(\varpi)^{\prime}\right)(x)<0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \forall x\in[x_{1},x_{3}],$ (3.4)

which is obviously true near the sonic point $P$. For future use we denote the
point $(x_{3},\varphi(x_{3}))$ by $R$ which lie on the curve $\widehat{PE}$.

###  3.3 Boundary data for $W$ and $Z$

The strategy of this article is to solve system (2.29) for $(W,Z)$ in a
partial hodograph plane and then return back to the solution via an inverse
transformation. Therefore we need to derive the boundary data for $(W,Z)$ on
the arc $\widehat{PR}$ using the functions $(\hat{\theta},\hat{\varpi})(x)$.

Now from (2.27) and noting that
$\tilde{\partial}_{0}=\dfrac{\tilde{\partial}_{+}+\tilde{\partial}_{-}}{2\cos\omega}$,
we have

$\displaystyle W+Z=-\dfrac{\tilde{\partial}_{0}\theta}{\varpi}.$ (3.5)

Similarly from (2.28), we have

$\displaystyle
W-Z=\dfrac{4a^{2}(\varpi)\sqrt{1-\varpi^{2}}\tilde{\partial}_{0}\varpi}{\varpi
F_{1}(\varpi)},$ (3.6)

which together with (3.5) implies

$\displaystyle W=\dfrac{2a^{2}(\varpi)\sqrt{1-\varpi^{2}}}{\varpi
F_{1}(\varpi)}\tilde{\partial}_{0}\varpi-\dfrac{1}{2\varpi}\tilde{\partial}_{0}\theta,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
Z=-\dfrac{2a^{2}(\varpi)\sqrt{1-\varpi^{2}}}{\varpi
F_{1}(\varpi)}\tilde{\partial}_{0}\varpi-\dfrac{1}{2\varpi}\tilde{\partial}_{0}\theta.$
(3.7)

Recalling that the curve $\widehat{PR}$ is a streamline, we have

$\displaystyle\tilde{\partial}_{0}\theta|_{\widehat{PR}}$
$\displaystyle=\varphi(x)\cos\hat{\theta}(x)\hat{\theta}^{\prime}(x)=\varphi(x)\dfrac{\cos\hat{\theta}(x)\varphi^{\prime\prime}(x)}{1+(\varphi^{\prime}(x))^{2}},$
$\displaystyle\tilde{\partial}_{0}\varpi|_{\widehat{PR}}$
$\displaystyle=\varphi(x)\cos\hat{\theta}(x)\hat{\varpi}^{\prime}(x),$

which combined with (3.7) yields

$\displaystyle\begin{cases}W|_{\widehat{PR}}=\dfrac{\varphi(x)\cos\hat{\theta}(x)}{2\hat{\varpi}}\bigg{[}\dfrac{4a^{2}(\hat{\varpi})\sqrt{1-\hat{\varpi}^{2}}}{F_{1}(\hat{\varpi})}\hat{\hat{\varpi}}^{\prime}(x)-\dfrac{\varphi^{\prime\prime}(x)}{1+(\varphi^{\prime}(x))^{2}}\bigg{]}:=\hat{b}(x),\\\
Z|_{\widehat{PR}}=-\dfrac{\varphi(x)\cos\hat{\theta}(x)}{2\hat{\varpi}}\bigg{[}\dfrac{4a^{2}(\hat{\varpi})\sqrt{1-\hat{\varpi}^{2}}}{F_{1}(\hat{\varpi})}\hat{\varpi}^{\prime}(x)+\dfrac{\varphi^{\prime\prime}(x)}{1+(\varphi^{\prime}(x))^{2}}\bigg{]}:=\hat{c}(x).\end{cases}$
(3.8)

For later use, we give the boundary data of $L=\tilde{\partial}_{0}I$ on
$\widehat{PR}$

$\displaystyle\tilde{\partial}_{0}I|_{\widehat{PR}}=\dfrac{\varphi(x)\cos\hat{\theta}(x)}{2\hat{\varpi}}\bigg{[}\dfrac{4a^{2}(\hat{\varpi})}{F_{1}(\hat{\varpi})}\hat{\varpi}^{\prime}(x)\bigg{]}:=-\hat{d}(x).$
(3.9)

Moreover, it suggests by the conditions (3.3) and (3.4) that

$\displaystyle\begin{cases}\hat{b}(x),\leavevmode\nobreak\
\hat{c}(x),\leavevmode\nobreak\ \hat{d}(x)\in C^{1}([x_{1},x_{3}]),\\\
\hat{m}_{0}\leq\hat{b}(x),\leavevmode\nobreak\ \hat{c}(x),\leavevmode\nobreak\
\hat{d}(x)\leq\hat{M}_{0},\leavevmode\nobreak\ \forall\leavevmode\nobreak\
x\in[x_{1},x_{3}]\end{cases}$ (3.10)

for some positive constants $\hat{m}_{0}$ and $\hat{M}_{0}$.

##  4 Existence and regularity of solution in partial hodograph plane

In this section, we solve the singular system (2.29) with the boundary data
(3.8) under the conditions (3.10) near the point $P$ by introducing a partial
hodograph transformation.

###  4.1 Reformulated problem in partial hodograph plane

We first reformulate the problem into a new problem by introducing a partial
hodograph transformation. We introduce the coordinate transformation
$(x,r)\longrightarrow(t,\psi)$ such that

$\displaystyle t=\cos\omega(x,r),\leavevmode\nobreak\ \leavevmode\nobreak\
\psi=\phi(x,r)-\phi_{1},$ (4.1)

where $\phi$ is the potential of irrotational relativistic flow such that
$\phi_{x}=u$ and $\phi_{r}=v$ with $\phi_{1}=\phi(x_{1})$.

From the transformation we can observe that $\varpi=\sqrt{1-t^{2}}$.
Therefore, we define $F_{1}(\varpi):=\hat{F}_{1}(t)$ where

$\displaystyle\hat{F}_{1}(t)=4a^{2}(t)(1-t^{2})+2i(t)p^{\prime\prime}(\rho(t))\gamma_{a(t)}^{2}f(a(t))>0.$
(4.2)

Then by using (2.28), we see that

$\displaystyle J$
$\displaystyle:=\dfrac{\partial(t,\psi)}{\partial(x,r)}=\begin{vmatrix}\dfrac{\partial
t}{\partial x}&\dfrac{\partial t}{\partial r}\vspace{0.2 cm}\\\
\dfrac{\partial\psi}{\partial x}&\dfrac{\partial\psi}{\partial
r}\end{vmatrix}$ $\displaystyle=\dfrac{a\gamma_{a}\cos\omega}{\gamma r\sin
2\omega}\dfrac{(\tilde{\partial}_{+}\varpi-\tilde{\partial}_{-}\varpi)}{\cos\omega}$
$\displaystyle J$
$\displaystyle=\dfrac{\gamma_{a}\hat{F}_{1}(t)(W+Z)}{4a(t)\gamma(t)rt}\neq
0\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
0\leq t<1.$ (4.3)

We next derive the boundary data of $\phi$ on $\widehat{PR}$. Noting the
definitions of $\tilde{\partial}_{i}\leavevmode\nobreak\ (i=0,\pm)$ it is easy
to obtain that

$\displaystyle\begin{cases}\tilde{\partial}_{0}\phi=r\cos\theta.\dfrac{a\gamma_{a}\cos\theta}{\gamma\sin\omega}+r\sin\theta.\dfrac{a\gamma_{a}\sin\theta}{\gamma\sin\omega}=\dfrac{a\gamma_{a}r}{\gamma\sin\omega},\\\
\tilde{\partial}_{+}\phi=r\cos\alpha.\dfrac{a\gamma_{a}\cos\theta}{\gamma\sin\omega}+r\sin\alpha.\dfrac{a\gamma_{a}\sin\theta}{\gamma\sin\omega}=\dfrac{a\gamma_{a}r\cos\omega}{\gamma\sin\omega},\\\
\tilde{\partial}_{-}\phi=r\cos\beta.\dfrac{a\gamma_{a}\cos\theta}{\gamma\sin\omega}+r\sin\beta.\dfrac{a\gamma_{a}\sin\theta}{\gamma\sin\omega}=\dfrac{a\gamma_{a}r\cos\omega}{\gamma\sin\omega}.\end{cases}$
(4.4)

Then noting that the curve $\widehat{PR}$ is a streamline and expression of
$\tilde{\partial}_{0}\phi$ in (4.4), one can obtain that

$\displaystyle\hat{\phi}^{\prime}(x)=\dfrac{\hat{a}(x)\hat{\gamma_{a}}(x)}{\hat{\gamma}(x)\hat{\varpi}(x)\cos\hat{\theta}(x)}=\dfrac{\hat{a}(x)\hat{\gamma_{a}}(x)\sqrt{1+\varphi^{\prime}(x)^{2}}}{\hat{\gamma}(x)\hat{\varpi}(x)}>0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ x\in[x_{1},x_{3}],$ (4.5)

where $\hat{\phi}(x)=\phi(x,\varphi(x))$. Then we obtain the boundary data of
$\hat{\phi}(x)$ on $\widehat{PR}$ as

$\displaystyle\phi|_{\widehat{PR}}=\hat{\phi}(x)=\phi_{1}+\displaystyle\int_{x_{1}}^{x}\dfrac{\hat{a}(s)\hat{\gamma_{a}}(s)\sqrt{1+\varphi^{\prime}(s)^{2}}}{\hat{\gamma}(s)\hat{\varpi}(s)}ds\leavevmode\nobreak\
\forall\leavevmode\nobreak\ x\in[x_{1},x_{3}].$ (4.6)

Now using the conditions that $\hat{\phi}^{\prime}>0$ and
$\hat{\varpi}^{\prime}<0$ it is easy to see that the curve
$\widehat{PR}:r=\varphi(x)$ in $x-r$ plane is transformed into a curve
$\widehat{P^{\prime}R^{\prime}}:\psi=\tilde{\psi}(t)(t\in[0,t_{0}])$ in the
half plane of $t\geq 0$ defined through a parameter $x$:

$\displaystyle t=\cos\hat{\omega}(x),\leavevmode\nobreak\ \leavevmode\nobreak\
\psi=\hat{\phi}(x)-\phi_{1},\leavevmode\nobreak\ \leavevmode\nobreak\
(x\in[x_{1},x_{3}])$ (4.7)

such that the number $t_{0}=\cos\hat{\omega}(x_{3})$ is a positive constant.
Moreover, since the function $\psi=\hat{\phi}(x)-\phi_{1}$ is strictly
increasing, there exists an inverse function $x$ such that
$x=\hat{x}(\psi)(\psi\in[0,\psi_{0}])$, where
$\psi_{0}=\hat{\phi}(x_{3})-\phi_{1}$. Now we denote
$\hat{b}(\psi)=\hat{b}(x(\psi)),\leavevmode\nobreak\
\hat{c}(\psi)=\hat{c}(x(\psi)),\leavevmode\nobreak\
\hat{d}(\psi)=\hat{d}(x(\psi))$. Then we obtain the boundary data of $(W,Z,L)$
on $\widehat{P^{\prime}R^{\prime}}$ such that

$\displaystyle
W|_{\widehat{P^{\prime}R^{\prime}}}=\hat{b}(\psi),\leavevmode\nobreak\
\leavevmode\nobreak\
Z|_{\widehat{P^{\prime}R^{\prime}}}=\hat{c}(\psi),\leavevmode\nobreak\
\leavevmode\nobreak\
L|_{\widehat{P^{\prime}R^{\prime}}}=-\hat{d}(\psi)\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ \psi\in[0,\psi_{0}].$ (4.8)

Then it is straightforward to see from (3.10) that

$\displaystyle\begin{cases}\hat{b}(\psi),\hat{c}(\psi),\hat{d}(\psi)\in
C^{1}[0,\psi_{0}],\\\ \hat{m}_{0}\leq\hat{b}(\psi),\leavevmode\nobreak\
\hat{c}(\psi),\leavevmode\nobreak\
\hat{d}(\psi)\leq\hat{M}_{0},\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ \psi\in[0,\psi_{0}]\end{cases}$ (4.9)

for some positive constants $\hat{m}_{0}$ and $\hat{M}_{0}$.

Further, in these new coordinates, the weighted normalized derivatives now
become

$\displaystyle\tilde{\partial}_{+}$
$\displaystyle=-\dfrac{2F(t)W}{t}\bigg{[}\dfrac{\partial}{\partial
t}-\dfrac{ar\gamma_{a}t^{2}}{2\gamma
F(t)W\sqrt{1-t^{2}}}\dfrac{\partial}{\partial\psi}\bigg{]},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tilde{\partial}_{-}$
$\displaystyle=\dfrac{2F(t)Z}{t}\bigg{[}\dfrac{\partial}{\partial
t}+\dfrac{ar\gamma_{a}t^{2}}{2\gamma
F(t)Z\sqrt{1-t^{2}}}\dfrac{\partial}{\partial\psi}\bigg{]}$ (4.10)

where $F(t)=\dfrac{(1-t^{2})\hat{F}_{1}(t)}{4\hat{a}^{2}(t)}>0.$

Then we derive the characteristic decompositions of $(W,Z)$ in the partial
hodograph $(t,\psi)$ plane. Using the normalized derivatives (4.10) in
$t-\psi$ plane in (2.29), it is easy to obtain the reformulated characteristic
decompositions of $(W,Z)$ of the form

$\displaystyle\begin{cases}&W_{t}+\dfrac{ar\gamma_{a}t^{2}}{2\gamma
F(t)Z\sqrt{1-t^{2}}}W_{\psi}=\bigg{[}1+\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\bigg{]}\dfrac{W}{ZF(t)}\left(\dfrac{W-Z}{2t}\right)\\\
&\hskip
56.9055pt+\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})+\dfrac{\sin\beta+4\sin\theta
t}{4Z}\bigg{]}\dfrac{Wt}{F(t)}-\dfrac{\sin\beta}{4F(t)Z}Zt+\dfrac{\sin^{2}\theta}{2F(t)Z}t,\\\
&Z_{t}-\dfrac{ar\gamma_{a}t^{2}}{2\gamma
F(t)W\sqrt{1-t^{2}}}Z_{\psi}=\bigg{[}1+\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\bigg{]}\dfrac{Z}{WF(t)}\left(\dfrac{Z-W}{2t}\right)\\\
&\hskip
56.9055pt+\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})-\dfrac{\sin\alpha+4\sin\theta
t}{4W}\bigg{]}\dfrac{Zt}{F(t)}+\dfrac{\sin\alpha}{4F(t)W}Wt+\dfrac{\sin^{2}\theta}{2F(t)W}t,\\\
\end{cases}$ (4.11)

It is easy to see that (4.11) is not a closed system because there are two
unknown functions $r(t,\psi)$ and $\theta(t,\psi)$ in the system. In order to
close the system, we need to add the characteristic equations and boundary
data of $r$ and $\theta$ to the system. First of all, from the definitions of
weighted directional derivatives, we have the characteristic decompositions of
the form:

$\displaystyle\tilde{\partial}_{+}r=r\sin\alpha,\leavevmode\nobreak\
\leavevmode\nobreak\ \tilde{\partial}_{-}r=r\sin\beta,$

from which one can have

$\displaystyle\partial^{-}r=-\dfrac{tr\sin\alpha}{2FW},\leavevmode\nobreak\
\leavevmode\nobreak\ \partial^{+}r=\dfrac{tr\sin\beta}{2FZ},$ (4.12)

where $\partial^{\pm}=\partial_{t}+\lambda_{\pm}\partial_{\psi}$ such that
$\lambda_{+}=\dfrac{ar\gamma_{a}t^{2}}{2\gamma F(t)Z\sqrt{1-t^{2}}}$ and
$\lambda_{-}=-\dfrac{ar\gamma_{a}t^{2}}{2\gamma F(t)W\sqrt{1-t^{2}}}$.

Similarly, from (2.27) we get the characteristic decomposition of $\theta$ of
the form

$\displaystyle\partial^{-}\theta=\dfrac{t\sqrt{1-t^{2}}(\sin\theta+2tW)}{2FW},\leavevmode\nobreak\
\leavevmode\nobreak\
\partial^{+}\theta=\dfrac{t\sqrt{1-t^{2}}(\sin\theta-2tZ)}{2FZ}.$ (4.13)

Also, the boundary data of $r$ and $\theta$ on
$\widehat{P^{\prime}R^{\prime}}$ are

$\displaystyle
r|_{\widehat{P^{\prime}R^{\prime}}}=\varphi(\hat{x}(\psi))=:\hat{\varphi}(\psi),\leavevmode\nobreak\
\leavevmode\nobreak\
\theta|_{\widehat{P^{\prime}R^{\prime}}}=\hat{\theta}(\hat{x}(\psi))=:\hat{\theta}(\psi),$
(4.14)

which satisfy

$\displaystyle\begin{cases}\hat{\varphi}(\psi),\hat{\theta}(\psi)\in
C^{1}([0,\psi_{0}]),\\\
\varphi_{0}\leq\hat{\varphi}(\psi)\leq\varphi_{1}\forall\leavevmode\nobreak\
\psi\in[0,\psi_{0}].\end{cases}$ (4.15)

Combining (4.11), (4.12) and (4.13), we obtain a closed system of
$(W,Z,r,\theta)(t,\psi)$ of the form

$\displaystyle\begin{cases}&\partial^{+}W=\bigg{[}1+\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\bigg{]}\dfrac{W}{ZF(t)}\left(\dfrac{W-Z}{2t}\right)+\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})+\dfrac{\sin\beta+4\sin\theta
t}{4Z}\bigg{]}\dfrac{Wt}{F(t)}\\\ &\hskip
113.81102pt-\dfrac{\sin\beta}{4F(t)Z}Zt+\dfrac{t\sin^{2}\theta}{2F(t)Z},\\\
&\partial^{-}Z=\bigg{[}1+\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\bigg{]}\dfrac{Z}{WF(t)}\left(\dfrac{Z-W}{2t}\right)+\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})-\dfrac{\sin\alpha+4\sin\theta
t}{4W}\bigg{]}\dfrac{Zt}{F(t)}\\\ &\hskip
113.81102pt+\dfrac{\sin\alpha}{4F(t)W}Wt+\dfrac{t\sin^{2}\theta}{2F(t)W},\\\
&\partial^{-}r=-\dfrac{tr\sin\alpha}{2FW},\vspace{0.2 cm}\\\
&\partial^{-}\theta=\dfrac{t\sqrt{1-t^{2}}(\sin\theta+2tW)}{2FW}\end{cases}$
(4.16)

with the boundary data (4.8) and (4.14). It is easy to observe that the two
eigenvalues of the system (4.16) are $\lambda_{\pm}$ defined as before.

In the upcoming subsections, we try to obtain the existence and regularity of
a solution $(W,Z,r,\theta)$ of the closed system $\eqref{eq: 3.45}$ with the
boundary data (4.8) and (4.14) in the neighbourhood
$P^{\prime}F^{\prime}D^{\prime}$ of the point $P^{\prime}$, where $F^{\prime}$
is a point $(\bar{\delta},\tilde{\psi}(\bar{\delta}))$ lying on the streamline
$\widehat{P^{\prime}R^{\prime}}$ such that $\bar{\delta}\in(0,t_{0}]$ is a
small positive number; see Figure 5.

###  4.2 A strong determinate domain and a priori estimates

In this subsection, we construct a strong determinate domain $\Omega$ for
system (4.16), which is not easy due to the nonlinearity of the system.

Noting the fact that $0<a^{2}<1,\leavevmode\nobreak\
p^{\prime\prime}(\rho)>0,\leavevmode\nobreak\ i(\rho)>0,\leavevmode\nobreak\
0<\gamma_{a}<1,\leavevmode\nobreak\ f(a)>0$, let us set

$\displaystyle\overline{K}=1+$
$\displaystyle\max\bigg{\\{}1+\left(\dfrac{2\hat{m}_{0}+1}{\hat{m}_{0}}\right)\bigg{[}\dfrac{2a^{2}}{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}\bigg{]},1+\left(\dfrac{2\hat{m}_{0}^{2}+2\hat{m}_{0}+2}{\hat{m}_{0}^{2}}\right)\bigg{[}\dfrac{2a^{2}}{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}\bigg{]}\bigg{\\}}<\infty.$
(4.17)

Further, we choose $\delta_{0}=\min\\{1/\sqrt{2},t_{0}\\}$ such that

$\displaystyle
F(t)=(1-t^{2})\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+1-t^{2}\bigg{]}\geq\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\geq\dfrac{\kappa}{2}>0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ t\in(0,\delta_{0}]$ (4.18)

for some $\kappa>0$. Moreover, we choose a positive number
$\delta_{1}\leq\delta_{0}$ such that

$\displaystyle e^{\overline{K}\delta_{1}^{2}}\leq 2,\leavevmode\nobreak\
\leavevmode\nobreak\ \delta_{1}\leq\dfrac{\kappa\hat{m}_{0}}{3}.$ (4.19)

We next derive the slope of curve
$\widehat{P^{\prime}R^{\prime}}\cap\\{t\leq\delta_{0}\\}$. From (4.5) and
(4.7) we have

$\displaystyle\tilde{\psi}^{\prime}(t)=-\hat{\phi}^{\prime}(x).\dfrac{\cos\hat{\omega}}{\hat{\varpi}(x)\hat{\varpi}^{\prime}(x)}=-\dfrac{\hat{a}\hat{\gamma}_{a}\sqrt{1+\varphi^{\prime
2}}}{\hat{\gamma}\hat{\varpi}^{\prime}\hat{\varpi}^{2}}t.$ (4.20)

Then if we denote
$\tilde{m}=\underset{-\psi\in[0,\tilde{\psi}(\delta_{0})]}{\min}\dfrac{\hat{a}\hat{\gamma}_{a}\sqrt{1+\varphi^{\prime
2}}}{\hat{\gamma}\hat{\varpi}^{\prime}\hat{\varpi}^{2}}(\hat{x}(\psi))>0$ and
set
$\widetilde{K}=\max\bigg{\\{}\dfrac{8\varphi_{1}}{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)\hat{m}_{0}}exp\left(\dfrac{2t_{0}^{2}}{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)\hat{m}_{0}}\right)\bigg{\\}}<\infty$
such that there exists a $\delta_{2}\leq\delta_{0}$ which satisfy
$\widetilde{K}\delta_{2}<\tilde{m}$.

Let $\delta=\min\\{\delta_{1},\delta_{2}\\}$. Then we consider the curve
$\psi=\bar{\psi}(t)$ defined by

$\displaystyle\bar{\psi}(t)=\tilde{\psi}(\delta)-\dfrac{\widetilde{K}}{3}\delta^{3}+\dfrac{\widetilde{K}}{3}t^{3}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ t\in[0,\delta].$ (4.21)

Then by integrating $\eqref{eq: 3.61}$, it is easy to see that

$\displaystyle\tilde{\psi}(\delta)\geq\displaystyle\int_{0}^{\delta}\widetilde{K}tdt=\dfrac{\tilde{m}}{2}\delta^{2}>\dfrac{\widetilde{K}\delta_{2}}{2}\delta^{2}>\dfrac{\widetilde{K}}{3}\delta^{3}.$
(4.22)

so that $\bar{\psi}(0)>0$.

Figure 5: The strong determinate domain $\Omega$

Let us denote the point $(0,\bar{\psi}(0))$ by $D^{\prime}$ and the point
$(\delta,\tilde{\psi}(\delta))$ by $T^{\prime}$. Further, let $\Omega$ be the
domain bounded by the curves $\widehat{P^{\prime}T^{\prime}}$,
$\widehat{D^{\prime}T^{\prime}}$ and the degenerate line
$\widehat{P^{\prime}D^{\prime}}$. Moreover, $\epsilon\in(0,\delta]$ be an
arbitrary constant such that we denote
$\Omega_{\epsilon}=\Omega\cap\\{(t,\psi)|t\geq\epsilon\\}$; see Figure 5. Then
we have the following Lemma:

####  4.2.1 Upper and lower bounds of $W,Z$ and $r$ in $t-\psi$ plane

######  Lemma 4.1.

Let us assume that the conditions (4.9) and (4.15) are satisfied such that
there exists a $C^{1}$ solution $(W,Z,r,\theta)(t,\psi)$ of the system (4.16)
with the boundary data (4.8) and (4.14) in the domain $\Omega_{\epsilon}$.
Then

$\displaystyle\begin{cases}\dfrac{\hat{m}_{0}}{2}<W(t,\psi),Z(t,\psi)<\hat{M}_{0}+1,\\\
\varphi_{0}exp\left(-\dfrac{t_{0}^{2}}{\kappa\hat{m}_{0}}\right)\leq
r(t,\psi)\leq\varphi_{1}exp\left(\dfrac{t_{0}^{2}}{\kappa\hat{m}_{0}}\right)\end{cases}$
(4.23)

for all $(t,\psi)\in\Omega_{\epsilon}$.

###### Proof.

In order to prove this Lemma, we need to prove that the region
$\mathbb{I}:=\left(\dfrac{\hat{m}_{0}}{2},\hat{M}_{0}+1\right)\times\left(\dfrac{\hat{m}_{0}}{2},\hat{M}_{0}+1\right)$
is an invariant region for $(W,Z)(t,\psi)$.

Let us consider the level curve
$t=\epsilon^{\prime}(\epsilon^{\prime}\in[\epsilon,\delta])$ and move it from
$t=\delta$ to $t=\epsilon$. We assume, on the contrary, that the region
$\mathbb{I}$ is not an invariant region and consider that the point $A$ is the
first time on one such level set $t=\epsilon^{\prime}$ such that one of $W$
and $Z$ touches the lower boundary of $\mathbb{I}$. Without loss of
generality, we assume that $W|_{A}=\dfrac{\hat{m}_{0}}{2}$ and
$(W,Z)\in\mathbb{I}$ for any
$(t,\psi)\in\Omega_{\epsilon}\cap\\{t>\epsilon^{\prime}\\}$. Now it is easy to
check by applying the third equation of (4.16) that

$\displaystyle\varphi_{0}exp\left(-\dfrac{t_{0}^{2}}{\kappa\hat{m}_{0}}\right)\leq
r(t,\psi)\leq\varphi_{1}exp\left(\dfrac{t_{0}^{2}}{\kappa\hat{m}_{0}}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\
(t,\psi)\in\Omega_{\epsilon}\cap\\{t>\epsilon^{\prime}\\}.$

Hence, we draw a positive characteristic $l_{1}^{+}$ from the point $A$ up to
a point $A_{1}$ on the boundary curve $\widehat{P^{\prime}T^{\prime}}$.
Further, we set

$\displaystyle\widetilde{W}=W-g(t),\leavevmode\nobreak\ \leavevmode\nobreak\
\widetilde{Z}=Z-g(t),$ (4.24)

where $\leavevmode\nobreak\ \leavevmode\nobreak\
g(t)=\dfrac{\hat{m}_{0}}{2}e^{\overline{K}t^{2}}.$

By the choice of $\delta_{1}$, we have $\widetilde{W}|_{A_{1}}>0.$ Also, it
follows by $W|_{A}=\hat{m}_{0}/2$ that $\widetilde{W}|_{A}<0$, which implies
by the continuity of $W$ and $g$ that there exists a point $B$ lying between
$A$ and $A_{1}$ on $l_{1}^{+}$ such that $\widetilde{W}|_{B}=0$ and
$\widetilde{W}|_{\widehat{BA_{1}}}\geq 0$. Therefore,
$\partial^{+}\widetilde{W}|_{B}\geq 0$.

However, using the equations of $W$ and $Z$ from the system (4.16) and
performing a direct calculation, one can obtain the following characteristic
decompositions of $(\widetilde{W},\widetilde{Z})$

$\displaystyle\begin{cases}&\partial^{+}\widetilde{W}=\bigg{[}1+\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\bigg{]}\dfrac{(\widetilde{W}+g)}{(\widetilde{Z}+g)F(t)}\left(\dfrac{\widetilde{W}-\widetilde{Z}}{2t}\right)+\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})+\dfrac{\sin\beta+4\sin\theta
t}{4(\widetilde{Z}+g)}\bigg{]}\dfrac{\widetilde{W}t}{F(t)}\\\ &\hskip
85.35826pt+\dfrac{\widetilde{Z}t}{F(t)(\widetilde{Z}+g)}\Upsilon_{1}+\dfrac{t}{F(t)(\widetilde{Z}+g)}\Upsilon_{2},\vspace{0.3
cm}\\\
&\partial^{-}\widetilde{Z}=\bigg{[}1+\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\bigg{]}\dfrac{(\widetilde{Z}+g)}{(\widetilde{W}+g)F(t)}\left(\dfrac{\widetilde{Z}-\widetilde{W}}{2t}\right)+\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})-\dfrac{\sin\alpha+4\sin\theta
t}{4(\widetilde{W}+g)}\bigg{]}\dfrac{\widetilde{Z}t}{F(t)}\\\ &\hskip
85.35826pt+\dfrac{\widetilde{W}t}{F(t)(\widetilde{W}+g)}\Upsilon_{3}+\dfrac{t}{F(t)(\widetilde{W}+g)}\Upsilon_{4},\\\
\end{cases}$ (4.25)

where

$\displaystyle\Upsilon_{1}$
$\displaystyle=\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}g(t)-2F(t)g(t)\overline{K}-\dfrac{\sin\beta}{4},$
$\displaystyle\Upsilon_{2}$
$\displaystyle=\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}g^{2}(t)-2F(t)g^{2}(t)\overline{K}+g(t)\sin\theta
t+\dfrac{\sin^{2}\theta}{2},$ $\displaystyle\Upsilon_{3}$
$\displaystyle=\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}g(t)-2F(t)g(t)\overline{K}+\dfrac{\sin\alpha}{4},$
$\displaystyle\Upsilon_{4}$
$\displaystyle=\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}g^{2}(t)-2F(t)g^{2}(t)\overline{K}-g(t)\sin\theta
t+\dfrac{\sin^{2}\theta}{2}.$

For $\Upsilon_{1}$ and $\Upsilon_{3}$, one has

$\displaystyle\Upsilon_{1},\leavevmode\nobreak\ \Upsilon_{3}$
$\displaystyle\leq\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}g(t)-2F(t)g(t)\overline{K}+\dfrac{1}{4},$
$\displaystyle\leq
g\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2+\dfrac{1}{4g}-2F(t)\overline{K}\bigg{]}$
$\displaystyle\leq
g(t)\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2+\dfrac{1}{\hat{m}_{0}}-\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}\overline{K}\bigg{]}$
$\displaystyle=\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}g(t)\left(1+\left(\dfrac{2\hat{m}_{0}+1}{\hat{m}_{0}}\right)\bigg{[}\dfrac{2a^{2}}{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}\bigg{]}-\overline{K}\right)<0$

by the choice of $\overline{K}$ in (4.17).

Similarly, for $\Upsilon_{2}$ and $\Upsilon_{4}$ also, we have

$\displaystyle\Upsilon_{2},$
$\displaystyle\Upsilon_{4}\leq\left(\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2\right)g^{2}-\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}g^{2}\overline{K}+g(t)+\dfrac{1}{2}$
$\displaystyle\leq\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}g^{2}(t)\left(1+\left(\dfrac{2\hat{m}_{0}^{2}+2\hat{m}_{0}+2}{\hat{m}_{0}^{2}}\right)\bigg{[}\dfrac{2a^{2}}{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}\bigg{]}-\overline{K}\right)<0$

by the choice of $\overline{K}$ in (4.17).

Therefore, using the equation of $\widetilde{W}$ from (4.25) and noting the
facts that $\Upsilon_{1},\Upsilon_{2}<0$, one can conclude that
$\partial^{+}\widetilde{W}|_{B}<0$, which leads to a contradiction.

Similarly, if there exists a point $B^{\prime}$ lying on a level curve
$t=\epsilon^{\prime}$ in $\Omega_{\epsilon}$ such that one of $W$ or $Z$
touches the upper boundary of $\mathbb{I}$. Again, without loss of generality,
we assume that $W|_{B^{\prime}}=\hat{M}_{0}+1$ and $(W,Z)\in\mathbb{I}$ for
any $(t,\psi)\in\Omega_{\epsilon}\cap\\{t>\epsilon^{\prime}\\}$. Thus, we can
draw a positive characteristic $l_{2}^{+}$ from $B^{\prime}$ up to a point
$B_{1}$ lying on the boundary curve $\widehat{P^{\prime}T^{\prime}}$. On the
curve $\widehat{B^{\prime}B_{1}}$, we have $W\leq\hat{M}_{0}+1$, which implies
that $\partial^{+}W|_{B^{\prime}}\leq 0$. However, using the equation for $W$
from (4.16), one can obtain

$\displaystyle\partial^{+}W|_{B^{\prime}}\geq\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})+\dfrac{\sin\theta
t}{Z}\bigg{]}\dfrac{Wt}{F(t)}+\dfrac{\sin^{2}\theta}{2F(t)Z}t>0,$

where we have used the fact
$\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}Z+t\sin\theta\geq\dfrac{\kappa}{2}\hat{m}_{0}-\delta|\sin\theta|>0$,
which is true by the choice of $\delta_{1}$.

The above two conclusions lead to a contradiction that proves that the region
$\mathbb{I}$ is an invariant region of $(W,Z)$. Furthermore, the estimates of
$r(t,\psi)$ can be easily obtained using the third equation of (4.16).
Therefore, the proof of the Lemma is completed. ∎

Now we consider a function class $\mathcal{S}(\overline{\Omega})$ which
incorporates all vector functions
$\mathbf{F}=(f_{1},f_{2},f_{3},f_{4})^{T}:\overline{\Omega}\longrightarrow\mathbb{R}^{4}$
satisfying the following properties:

$\displaystyle\begin{cases}(P_{1}):\leavevmode\nobreak\ f_{1},f_{2}\in
C^{1}(\overline{\Omega}/\\{t=0\\}),f_{3},f_{4}\in C^{1}(\overline{\Omega});\\\
(P_{2}):\leavevmode\nobreak\
(f_{1},f_{2},f_{3},f_{4})^{T}(t,\tilde{\psi}(t))=(\hat{b}(\tilde{\psi}(t)),\hat{c}(\tilde{\psi}(t)),\hat{\varphi}(\tilde{\psi}(t)),\hat{\theta}(\tilde{\psi}(t)))^{T}\leavevmode\nobreak\
\forall t\in[0,\delta];\\\ (P_{3}):\leavevmode\nobreak\
\dfrac{\hat{m}_{0}}{2}<f_{1},f_{2}<\hat{M}_{0}+1,\leavevmode\nobreak\
\leavevmode\nobreak\
\varphi_{0}exp\left(-\dfrac{t_{0}^{2}}{\kappa\hat{m}_{0}}\right)\leq
f_{3}\leq\varphi_{1}exp\left(\dfrac{t_{0}^{2}}{\kappa\hat{m}_{0}}\right).\end{cases}$
(4.26)

It can be easily observed from Lemma 4.1 that $\mathcal{S}(\overline{\Omega})$
is not empty. Further, based on the expression of $\lambda_{+}$, we define the
curve $\psi=\psi_{+}(t;\xi,\eta)$ by

$\displaystyle\begin{cases}\dfrac{d\psi_{+}(t;\xi,\eta)}{dt}=\dfrac{ra\gamma_{a}t^{2}}{2\gamma
FZ\sqrt{1-t^{2}}}(t;\xi,\eta),\\\ \psi_{+}(\xi;\xi,\eta)=\eta,\end{cases}$
(4.27)

for $t\geq\xi$, where $(\xi,\eta)$ is an arbitrary point in $\Omega$. Then we
proceed to prove that $\Omega$ is a strong determinate domain in the following
Lemma:

######  Lemma 4.2.

If the solution $(W,Z,r,\theta)(t,\psi)$ of the system (4.16) with the
boundary data (4.8) and (4.14) belongs to the function class
$\mathcal{S}(\overline{\Omega})$ then $\Omega$ is a strong determinate domain.

###### Proof.

In order to prove this Lemma, it is enough to prove that the curve
$\psi=\psi_{+}(t;\xi,\eta)$ intersects only with the curve
$\widehat{P^{\prime}T^{\prime}}$. We prove this by proving that the slope of
curve $\psi=\psi_{+}(t)$ is strictly smaller than $\psi=\bar{\psi}(t)$ at any
point on $\widehat{D^{\prime}T^{\prime}}$. Indeed, by using (4.17)

$\displaystyle\dfrac{ra\gamma_{a}t^{2}}{2\gamma
FZ\sqrt{1-t^{2}}}\leq\dfrac{t^{2}}{2.\dfrac{\kappa}{2}\dfrac{\hat{m}_{0}}{2}.\dfrac{1}{\sqrt{2}}}\varphi_{1}exp\left(\dfrac{t_{0}^{2}}{\kappa\hat{m}_{0}}\right)<\widetilde{K}t^{2}=\bar{\psi}^{\prime}(t)$
(4.28)

for $t\in(0,\delta)$ which implies that the domain $\Omega$ is a strong
determinate domain. ∎

###  4.3 Existence of Solutions

To establish the existence of solutions, we first need to derive a priori
$C^{1}$ estimates. For this purpose, we first introduce
$\overline{W}=\dfrac{1}{W},\leavevmode\nobreak\ \overline{Z}=\dfrac{1}{Z}$ to
convert the system (4.16) into the following form

$\displaystyle\begin{cases}\bar{\partial}^{+}\overline{W}=\dfrac{\overline{W}-\overline{Z}}{2t}+tH_{1}(\overline{W},\overline{Z},\theta,t),\vspace{0.2
cm}\\\
\bar{\partial}^{-}\overline{Z}=\dfrac{\overline{Z}-\overline{W}}{2t}+tH_{2}(\overline{W},\overline{Z},\theta,t),\vspace{0.2
cm}\\\ \bar{\partial}^{-}r=-\dfrac{\overline{W}tr\sin\alpha}{2F},\vspace{0.2
cm}\\\
\bar{\partial}^{-}\theta=\dfrac{t\sqrt{1-t^{2}}(\overline{W}\sin\theta+2t)}{2F},\end{cases}$
(4.29)

where $\bar{\partial}^{\pm}=\partial_{t}+\bar{\lambda}_{\pm}\partial_{\psi}$
with $\bar{\lambda}_{+}=\dfrac{\overline{Z}ra\gamma_{a}}{2\gamma
F\sqrt{1-t^{2}}}t^{2}$,
$\bar{\lambda}_{-}=-\dfrac{\overline{W}ra\gamma_{a}}{2\gamma
F\sqrt{1-t^{2}}}t^{2}$,
and

$\displaystyle H_{1}=$
$\displaystyle\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2-t^{2}\bigg{]}\dfrac{(\overline{W}-\overline{Z})}{2F(t)}-\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}\dfrac{\overline{W}}{F(t)}$
$\displaystyle-\dfrac{\overline{W}}{2F}\bigg{\\{}\dfrac{\sin\beta}{2}(\overline{Z}-\overline{W})+2t\overline{Z}\sin\theta+\overline{W}\cdot\overline{Z}\sin^{2}\theta\bigg{\\}},$
$\displaystyle H_{2}=$
$\displaystyle\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2-t^{2}\bigg{]}\dfrac{(\overline{Z}-\overline{W})}{2F(t)}-\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\bigg{]}\dfrac{\overline{Z}}{F(t)}$
$\displaystyle-\dfrac{\overline{Z}}{2F}\bigg{\\{}\dfrac{\sin\alpha}{2}(\overline{Z}-\overline{W})-2t\overline{W}\sin\theta+\overline{W}\cdot\overline{Z}\sin^{2}\theta\bigg{\\}},$

We now use the following commutator relation

$\displaystyle\bar{\partial}^{-}\bar{\partial}^{+}-\bar{\partial}^{+}\bar{\partial}^{-}=\dfrac{\bar{\partial}^{-}\bar{\lambda}_{+}-\bar{\partial}^{+}\bar{\lambda}_{-}}{\bar{\lambda}_{+}-\bar{\lambda}_{-}}(\bar{\partial}^{+}-\bar{\partial}^{-})$
(4.30)

to obtain the equations of $\bar{\partial}^{-}\overline{W}$ and
$\bar{\partial}^{+}\overline{Z}$ as follows:

$\displaystyle\begin{cases}\bar{\partial}^{+}\bar{\partial}^{-}\overline{W}=\bar{\partial}^{-}\bar{\partial}^{+}\overline{W}+\dfrac{\bar{\partial}^{-}\bar{\lambda}_{+}-\bar{\partial}^{+}\bar{\lambda}_{-}}{\bar{\lambda}_{+}-\bar{\lambda}_{-}}(\bar{\partial}^{-}\overline{W}-\bar{\partial}^{+}\overline{W}),\\\
\bar{\partial}^{-}\bar{\partial}^{+}\overline{Z}=\bar{\partial}^{+}\bar{\partial}^{-}\overline{Z}+\dfrac{\bar{\partial}^{-}\bar{\lambda}_{+}-\bar{\partial}^{+}\bar{\lambda}_{-}}{\bar{\lambda}_{+}-\bar{\lambda}_{-}}(\bar{\partial}^{+}\overline{Z}-\bar{\partial}^{-}\overline{Z}).\end{cases}$
(4.31)

A routine calculation now yields

$\displaystyle\dfrac{\bar{\partial}^{-}\bar{\lambda}_{+}-\bar{\partial}^{+}\bar{\lambda}_{-}}{\bar{\lambda}_{+}-\bar{\lambda}_{-}}=\dfrac{2}{t}+th,$
(4.32)

where

$\displaystyle h=$
$\displaystyle\bigg{[}\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+4(1-t^{2})\bigg{]}\dfrac{1}{F(t)}+\dfrac{1}{1-t^{2}}+\dfrac{H_{1}+H_{2}}{\overline{W}+\overline{Z}}-\dfrac{\overline{W}\cdot\overline{Z}\sqrt{1-t^{2}}\cos\theta}{\overline{W}+\overline{Z}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\dfrac{i\gamma_{a}^{2}t(1-t^{2})}{F^{2}}\left(\gamma_{a}^{2}p^{\prime\prime}(\rho)+\tau^{\prime}(\rho)\right),$

where $\tau=\dfrac{i\gamma_{a}^{4}p^{\prime\prime}(\rho)}{2a^{2}}$ such that
$\tau^{\prime}$ is bounded in the domain $\Omega$.

Moreover, making use of (4.29), we find that

$\displaystyle\bar{\partial}^{-}\bar{\partial}^{+}\overline{W}=\dfrac{\bar{\partial}^{-}\overline{W}-\bar{\partial}^{-}\overline{Z}}{2t}-\dfrac{\overline{W}-\overline{Z}}{2t^{2}}+H_{1}+tf_{1}\bar{\partial}^{-}\overline{W}+tf_{2}\bar{\partial}^{-}\overline{Z}+tf_{3}\bar{\partial}^{-}\theta+tf_{4},$
(4.33)

where

$\displaystyle f_{1}$ $\displaystyle=\dfrac{\partial
H_{1}}{\partial\overline{W}}=\dfrac{1}{2F}\Bigg{\\{}3t^{2}-2-\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}-2t\overline{Z}\sin\theta-\dfrac{\sin\beta}{2}\overline{Z}-2\overline{W}.\overline{Z}\sin^{2}\theta+\sin\beta\overline{W}\Bigg{\\}},$
$\displaystyle f_{2}$ $\displaystyle=\dfrac{\partial
H_{1}}{\partial\overline{Z}}=\dfrac{1}{2F}\Bigg{\\{}t^{2}-2-\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}-2t\overline{W}\sin\theta-\dfrac{\sin\beta}{2}\overline{W}-\overline{W}^{2}\sin^{2}\theta\Bigg{\\}},$
$\displaystyle f_{3}$ $\displaystyle=\dfrac{\partial
H_{1}}{\partial\theta}=-\dfrac{\overline{W}}{2F}\Bigg{\\{}2t\overline{Z}\cos\theta+\dfrac{\cos\beta}{2}(\overline{Z}-\overline{W})+\overline{W}.\overline{Z}\sin
2\theta\Bigg{\\}},$ $\displaystyle f_{4}$ $\displaystyle=\dfrac{\partial
H_{1}}{\partial
t}=\dfrac{H_{1}}{F}\Bigg{\\{}\bigg{[}\left(\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\right)2t-i\gamma_{a}^{2}t^{2}(1-t^{2})\left(\gamma_{a}^{2}p^{\prime\prime}(\rho)+\tau^{\prime}(\rho)\right)\bigg{]}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+3t\overline{W}+t\overline{Z}-\overline{W}.\overline{Z}\sin\theta+i\gamma_{a}^{2}t^{2}\left(\gamma_{a}^{2}p^{\prime\prime}(\rho)+\tau^{\prime}(\rho)\right)\left(\dfrac{\overline{W}+\overline{Z}}{2}\right)\Bigg{\\}}$

and

$\displaystyle\bar{\partial}^{+}\bar{\partial}^{-}\overline{Z}=\dfrac{\bar{\partial}^{+}\overline{Z}-\bar{\partial}^{+}\overline{W}}{2t}-\dfrac{\overline{Z}-\overline{X}}{2t^{2}}+H_{2}+tg_{1}\bar{\partial}^{+}\overline{Z}+tg_{2}\bar{\partial}^{+}\overline{W}+tg_{3}\bar{\partial}^{+}\theta+tg_{4},$
(4.34)

where

$\displaystyle g_{1}$ $\displaystyle=\dfrac{\partial
H_{2}}{\partial\overline{Z}}=\dfrac{1}{2F}\Bigg{\\{}3t^{2}-2-\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2t\overline{W}\sin\theta+\dfrac{\sin\alpha}{2}\overline{W}-2\overline{W}.\overline{Z}\sin^{2}\theta-\sin\alpha\overline{Z}\Bigg{\\}},$
$\displaystyle g_{2}$ $\displaystyle=\dfrac{\partial
H_{2}}{\partial\overline{W}}=\dfrac{1}{2F}\Bigg{\\{}t^{2}-2-\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2t\overline{Z}\sin\theta+\dfrac{\sin\alpha}{2}\overline{Z}-\overline{Z}^{2}\sin^{2}\theta\Bigg{\\}},$
$\displaystyle g_{3}$ $\displaystyle=\dfrac{\partial
H_{2}}{\partial\theta}=-\dfrac{\overline{Z}}{2F}\Bigg{\\{}-2t\overline{W}\cos\theta+\dfrac{\cos\alpha}{2}(\overline{Z}-\overline{W})+\overline{W}.\overline{Z}\sin
2\theta\Bigg{\\}},$ $\displaystyle g_{4}$ $\displaystyle=\dfrac{\partial
H_{2}}{\partial
t}=\dfrac{H_{2}}{F}\Bigg{\\{}\bigg{[}\left(\dfrac{i\gamma_{a}^{2}p^{\prime\prime}(\rho)f(a)}{2a^{2}}+2(1-t^{2})\right)2t-i\gamma_{a}^{2}t^{2}(1-t^{2})\left(\gamma_{a}^{2}p^{\prime\prime}(\rho)+\tau^{\prime}(\rho)\right)\bigg{]}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+3t\overline{Z}+t\overline{W}+\overline{W}.\overline{Z}\sin\theta+i\gamma_{a}^{2}t^{2}\left(\gamma_{a}^{2}p^{\prime\prime}(\rho)+\tau^{\prime}(\rho)\right)\left(\dfrac{\overline{W}+\overline{Z}}{2}\right)\Bigg{\\}}.$

Therefore, inserting (4.32)-(4.34) into (4.31) and using (4.13), (4.29) one
can compute

$\displaystyle\begin{cases}\bar{\partial}^{+}\bar{\partial}^{-}\overline{W}=\left(\dfrac{5}{2t}+th+tf_{1}\right)\bar{\partial}^{-}\overline{W}+G_{1},\vspace{0.2
cm}\\\
\bar{\partial}^{-}\bar{\partial}^{+}\overline{Z}=\left(\dfrac{5}{2t}+th+tg_{1}\right)\bar{\partial}^{+}\overline{Z}+G_{2},\end{cases}$
(4.35)

where

$\displaystyle G_{1}$
$\displaystyle=\dfrac{5(\overline{Z}-\overline{W})}{4t^{2}}+\dfrac{h+f_{2}}{2}(\overline{Z}-\overline{W})-\dfrac{2H_{1}+H_{2}}{2}+\dfrac{t^{2}f_{3}\sqrt{1-t^{2}}(\overline{W}\sin\theta+2t)}{2F}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+tf_{4}+(f_{2}H_{2}-hH_{1})t^{2},$ $\displaystyle G_{2}$
$\displaystyle=\dfrac{5(\overline{W}-\overline{Z})}{4t^{2}}+\dfrac{h+g_{2}}{2}(\overline{W}-\overline{Z})-\dfrac{H_{1}+2H_{2}}{2}+\dfrac{t^{2}g_{3}\sqrt{1-t^{2}}(\overline{Z}\sin\theta-2t)}{2F}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+tg_{4}+(g_{2}H_{1}-hH_{2})t^{2}.$

We now employ the second order decompositions (4.35) to develop the $C^{1}$
estimates of solutions in the following Lemma:

######  Lemma 4.3.

Let us assume that the conditions (4.9) and (4.15) are satisfied such that
there exists a $C^{1}$ solution $(W,Z,r,\theta)(t,\psi)$ of the system (4.16)
with the boundary data (4.8) and (4.14) in the domain $\Omega_{\epsilon}$.
Then

$\displaystyle||(W,Z,r,\theta)||_{C^{1}(\Omega_{\epsilon})}\leq\dfrac{K_{1}}{\epsilon^{3}},$
(4.36)

where $K_{1}$ is a positive constant, independent of $\epsilon$.

###### Proof.

From Lemma 4.1, we know that the functions $W$ and $Z$ are bounded. Therefore,
in order to prove this Lemma, it is enough to prove that

$\displaystyle||(\overline{W},\overline{Z},r,\theta)||_{C^{1}(\Omega_{\epsilon})}\leq\dfrac{K_{1}}{\epsilon^{3}},$
(4.37)

To prove this, we use the first order characteristic decomposition (4.29) and
Lemma 4.1 to conclude that

$\displaystyle|\bar{\partial}^{+}\overline{W}|,\leavevmode\nobreak\
|\bar{\partial}^{-}\overline{Z}|\leq\dfrac{K_{1}}{t}\leq\dfrac{K_{1}}{\epsilon}$
(4.38)

for some positive constant $K_{1}$, independent of $\epsilon$. Also, by Lemma
(4.1) it is easy to observe that the coefficients of
$\bar{\partial}^{-}\overline{W}$ and $\bar{\partial}^{+}\overline{Z}$ and the
functions $G_{1}$ and $G_{2}$ in (4.35) satisfy

$\displaystyle\left|\dfrac{5}{2t}+th+tf_{1}\right|,\leavevmode\nobreak\
\left|\dfrac{5}{2t}+th+tg_{1}\right|\leq\dfrac{K_{1}}{t},\leavevmode\nobreak\
|G_{1}|,\leavevmode\nobreak\ |G_{2}|\leq\dfrac{K_{1}}{t^{2}}.$

Hence, one can integrate the second order decompositions (4.35) along the
positive and negative characteristic curves to obtain

$\displaystyle|\bar{\partial}^{-}\overline{W}|,\leavevmode\nobreak\
|\bar{\partial}^{+}\overline{Z}|\leq\dfrac{K_{1}}{\epsilon}$ (4.39)

for some positive constant $K_{1}$, independent of $\epsilon$. Therefore,
noting the expression of scaled normalized derivatives $\bar{\partial}^{\pm}$,
we obtain

$\displaystyle\partial_{t}=\dfrac{\overline{W}\bar{\partial}^{+}+\overline{Z}\bar{\partial}^{-}}{\overline{W}+\overline{Z}},\leavevmode\nobreak\
\leavevmode\nobreak\
\partial_{\psi}=\dfrac{2F\gamma\sqrt{1-t^{2}}}{ra\gamma_{a}(\overline{W}+\overline{Z})}.\dfrac{\bar{\partial}^{+}-\bar{\partial}^{-}}{t^{2}},$

which combined with (4.38) and (4.39) yields the $C^{1}$ a priori estimates of
$(\overline{W},\overline{Z})$ of the form

$\displaystyle||(\overline{W},\overline{Z})||_{C^{1}(\Omega_{\epsilon})}\leq\dfrac{K_{1}}{\epsilon^{3}}.$
(4.40)

The estimates of $(r,\theta)$ can be easily derived using the first order
decompositions of $r$ and $\theta$ and thus, the proof of the Lemma is
completed. ∎

The existence of a local $C^{1}$ solution in the neighborhood of the point
$T^{\prime}(\delta,\tilde{\psi}(\delta))$ can be obtained by using the
classical local existence results for boundary value problems to the system of
strictly hyperbolic equations [52]. Furthermore, utilizing the Lemmas 4.1 and
4.3 we can establish the existence of a global $C^{1}$ solution for the system
(4.16) with the boundary data (4.8) and (4.14) in the domain
$\Omega_{\epsilon}$ by the classical approach of extending local solution to a
larger domain by taking the level sets of $t$ as the Cauchy supports for any
$\epsilon>0$. It is noteworthy to see from the system (4.16) that the
extension step size depends only on the boundary data and the $C^{0},C^{1}$
norms of $(W,Z,r,\theta)$ which are uniformly bounded in $\Omega_{\epsilon}$.
Since the domain $\Omega_{\epsilon}$ is compact, the extension process can be
completed in a finite number of steps. Therefore, by the arbitrariness of
$\epsilon$, one can achieve the $C^{1}$ solution in $\Omega/\\{t=0\\}$.

###  4.4 Regularity of solutions

In this subsection, we explore the uniform regularity of solutions up to the
degenerate line $t=0$ in the partial hodograph plane. We first derive the
uniform boundedness of $L(t,\psi)$ which is uniformly bounded on the boundary
curve $\widehat{P^{\prime}T^{\prime}}$ by (4.1). Also, noting that
$\overline{W}=\dfrac{1}{W}$ and $\overline{Z}=\dfrac{1}{Z}$, we can express
$L$ as

$\displaystyle L=\dfrac{WZ}{2}.\dfrac{\overline{Z}-\overline{W}}{t}.$ (4.41)

Therefore, in order to prove the uniform boundedness of $L$, it is enough to
prove that the function $\overline{L}=(\overline{W}-\overline{Z})/t$ is
uniformly bounded up to the degenerate line $\widehat{P^{\prime}D^{\prime}}$,
i.e. $t=0$. A straightforward calculation using (4.29) leads us to the
equation of $\overline{L}$ of the form

$\displaystyle\begin{cases}\bar{\partial}^{+}\overline{L}=(H_{1}-H_{2})-\dfrac{\bar{\partial}^{+}\overline{Z}-\bar{\partial}^{-}\overline{Z}}{t},\\\
\bar{\partial}^{-}\overline{L}=(H_{1}-H_{2})-\dfrac{\bar{\partial}^{+}\overline{W}-\bar{\partial}^{-}\overline{W}}{t}\end{cases}$
(4.42)

where $H_{1}$ and $H_{2}$ are already defined in (4.29).

Therefore, noting the uniform boundedness of $H_{1}$ and $H_{2}$ it is
necessary to establish the estimates for
$(\bar{\partial}^{+}\overline{W}-\bar{\partial}^{-}\overline{W})/t$ and
$(\bar{\partial}^{+}\overline{Z}-\bar{\partial}^{-}\overline{Z})/t$.

Let us set

$\displaystyle
U=\bar{\partial}^{+}\overline{W}-\bar{\partial}^{-}\overline{W},\leavevmode\nobreak\
\leavevmode\nobreak\
V=\bar{\partial}^{+}\overline{Z}-\bar{\partial}^{-}\overline{Z}.$ (4.43)

Then one can use the commutator relation (4.30) again to obtain the
decompositions of $U$ and $V$ of the form

$\displaystyle\bar{\partial}^{+}U=\dfrac{\bar{\partial}^{-}\bar{\lambda}_{+}-\bar{\partial}^{+}\bar{\lambda}_{-}}{\bar{\lambda}_{+}-\bar{\lambda}_{-}}U+(\bar{\lambda}_{+}-\bar{\lambda}_{-})(\bar{\partial}^{+}\overline{W})_{\psi}$
(4.44)

and

$\displaystyle\bar{\partial}^{-}V=\dfrac{\bar{\partial}^{-}\bar{\lambda}_{+}-\bar{\partial}^{+}\bar{\lambda}_{-}}{\bar{\lambda}_{+}-\bar{\lambda}_{-}}V+(\bar{\lambda}_{+}-\bar{\lambda}_{-})(\bar{\partial}^{-}\overline{Z})_{\psi}.$
(4.45)

Using the same arguments as in the derivation of (4.35), we have

$\displaystyle\begin{cases}\bar{\partial}^{+}U=\left(\dfrac{5}{2t}+t\bar{f}_{1}\right)U+(2t^{2}\bar{f}_{2}-1)\dfrac{V}{2t}+t^{2}\bar{f}_{3},\vspace{0.2
cm}\\\
\bar{\partial}^{-}V=\left(\dfrac{5}{2t}+t\bar{g}_{1}\right)V+(2t^{2}\bar{g}_{2}-1)\dfrac{U}{2t}+t^{2}\bar{g}_{3},\end{cases}$
(4.46)

where $\bar{f}_{1}=f_{1}+h,\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{f}_{2}=f_{2},\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{f}_{3}=\dfrac{f_{3}\sqrt{1-t^{2}}}{2F}[\sin\theta(\overline{Z}-\overline{W})-4t],\leavevmode\nobreak\
\leavevmode\nobreak\ \bar{g}_{1}=g_{1}+h,\leavevmode\nobreak\
\leavevmode\nobreak\ \bar{g}_{2}=g_{2},\leavevmode\nobreak\
\leavevmode\nobreak\
\bar{g}_{3}=\dfrac{g_{3}\sqrt{1-t^{2}}}{2F}[\sin\theta(\overline{Z}-\overline{W})-4t]$.

Noting Lemma 4.1 and the uniform boundedness of $f_{i}$ and
$g_{i}\leavevmode\nobreak\ (i=1,2,3)$, it is easy to observe that the
functions $\bar{f}_{i},\bar{g}_{i},(i=1,2,3)$ are uniformly bounded in
$\Omega$. We further denote

$\displaystyle\overline{U}=\dfrac{\bar{\partial}^{+}\overline{W}-\bar{\partial}^{-}\overline{W}}{t^{\nu}}=\dfrac{U}{t^{\nu}},\leavevmode\nobreak\
\leavevmode\nobreak\
\overline{V}=\dfrac{\bar{\partial}^{+}\overline{Z}-\bar{\partial}^{-}\overline{Z}}{t^{\nu}}=\dfrac{V}{t^{\nu}}$
(4.47)

for some $\nu\in(0,2]$.

Then (4.46) yields

$\displaystyle\begin{cases}\bar{\partial}^{+}(t^{-\frac{2\nu+1}{2}}\overline{U})=\left(t^{2}\bar{f}_{2}-\dfrac{1}{2}\right)t^{-\frac{2\nu+3}{2}}\overline{V}+(t^{2}\bar{f}_{1}\overline{U}+t^{2}\bar{f}_{3})t^{-\frac{2\nu+3}{2}},\\\
\bar{\partial}^{-}(t^{-\frac{2\nu+1}{2}}\overline{V})=\left(t^{2}\bar{g}_{2}-\dfrac{1}{2}\right)t^{-\frac{2\nu+3}{2}}\overline{U}+(t^{2}\bar{g}_{1}\overline{V}+t^{2}\bar{g}_{3})t^{-\frac{2\nu+3}{2}}.\end{cases}$
(4.48)

We now use (4.48) to prove the uniform boundedness of $\overline{U}$ and
$\overline{V}$. Let $A^{\prime}(0,\psi_{1})$ and
$A^{\prime\prime}(0,\psi_{2})$ be any two fixed points on the degenerate line
$\widehat{P^{\prime}D^{\prime}}$ satisfying $\psi_{1}>\psi_{2}$. Then from the
points $A^{\prime}$ and $A^{\prime\prime}$, we draw positive and negative
characteristic curves up to the boundary $\widehat{P^{\prime}T^{\prime}}$ at
$A_{1}$ and $A_{2}$, respectively. Let
$\mathcal{D}(\psi_{1},\psi_{2})\subseteq\Omega$ be the region bounded by the
curves $\widehat{A^{\prime}A^{\prime\prime}},\leavevmode\nobreak\
\widehat{A^{\prime}A_{1}},\leavevmode\nobreak\ \leavevmode\nobreak\
\widehat{A^{\prime\prime}A_{2}}$ and $\widehat{A_{1}A_{2}}$. Further, let us
denote

$\displaystyle
M^{\prime}=\underset{\mathcal{D}(\psi_{1},\psi_{2})}{\max}\\{|\bar{f}_{1}|,|\bar{g}_{1}|,|\bar{f}_{2}|,|\bar{g}_{2}|,|\bar{f}_{3}|,|\bar{g}_{3}|\\}.$

Figure 6: The regions $P^{\prime}R^{\prime}D^{\prime}$, $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$

We then divide the region $\mathcal{D}(\psi_{1},\psi_{2})$ into two parts,
$\mathcal{D}_{1}:=\mathcal{D}(\psi_{1},\psi_{2})\cap\\{t<\bar{\epsilon}\\}$
and
$\mathcal{D}_{2}:=\mathcal{D}(\psi_{1},\psi_{2})\cap\\{t\geq\bar{\epsilon}\\}$
(see Figure 6), where $\bar{\epsilon}$ is a small number in $(0,\delta]$
satisfying $16\bar{\epsilon}^{2}M^{\prime}\leq 1$. Then by the choice of
$\bar{\epsilon}$, one easily observe that

$\displaystyle\dfrac{1}{2}+t^{2}|\bar{f}_{2}|\leq\dfrac{9}{16},\leavevmode\nobreak\
\leavevmode\nobreak\ t^{2}|\bar{f}_{1}|,\leavevmode\nobreak\
t^{2}|\bar{g}_{1}|,\leavevmode\nobreak\
t^{2}|\bar{f}_{3}|,\leavevmode\nobreak\
t^{2}|\bar{g}_{3}|\leq\dfrac{1}{16}\leavevmode\nobreak\
\forall(t,\psi)\in\overline{\mathcal{D}_{1}}.$ (4.49)

It is easy to observe that we need to check the uniform boundedness of
$\overline{U}$ and $\overline{V}$ in the region $\mathcal{D}_{1}$ only since
the degenerate line $t=0$ lies in $\mathcal{D}_{1}$ and $\overline{U}$ and
$\overline{V}$ are uniformly bounded in $\mathcal{D}_{2}$ and on the boundary
$\widehat{A_{1}A_{2}}$. Therefore, we now denote

$\displaystyle\widetilde{K}=1+3\max\\{\underset{\overline{\mathcal{D}_{2}}}{\max}\\{|\overline{U}|,|\overline{V}|\\},\underset{\widehat{A_{1}A_{2}}}{\max}\\{|\overline{U}|,|\overline{V}|\\}\\}.$
(4.50)

Then we have the following Lemma:

######  Lemma 4.4.

For any point $(t,\psi)\in\mathcal{D}_{1}$, there holds

$\displaystyle|\overline{U}|<\widetilde{K},\leavevmode\nobreak\
\leavevmode\nobreak\ |\overline{V}|<\widetilde{K}.$ (4.51)

for any $\nu\in(0,2]$.

###### Proof.

We prove this Lemma by the method of contradiction. Let $B(t_{b},\psi_{b})$ be
any point in $\mathcal{D}_{1}$ such that point $B$ is the first time that one
of $|\overline{U}|$ and $|\overline{V}|$ touches the boundary of
$(-\widetilde{K},\widetilde{K})\times(-\widetilde{K},\widetilde{K})$. Then
from point $B$, we draw a positive and negative characteristic curves up to
the upper boundary of $\mathcal{D}_{1}$ at points
$B_{1}(t_{b_{1}},\psi_{b_{1}})$ and $B_{2}(t_{b_{2}},\psi_{b_{2}})$,
respectively. Without loss of generality, we suppose that
$|\overline{U}(t_{b},\psi_{b})|=\widetilde{K}$ and
$|\overline{U}(t,\psi)|\leq\widetilde{K},\leavevmode\nobreak\
|\overline{V}(t,\psi)|\leq\widetilde{K}$ hold on the positive characteristic
curve $\widehat{BB_{1}}$. Further, in view of the choice of $\widetilde{K}$,
we see

$\displaystyle|\overline{U}(t_{b_{1}},\psi_{b_{1}})|,\leavevmode\nobreak\
|\overline{V}(t_{b_{1}},\psi_{b_{1}})|,\leavevmode\nobreak\
|\overline{U}(t_{b_{2}},\psi_{b_{2}})|,\leavevmode\nobreak\
|\overline{V}(t_{b_{2}},\psi_{b_{2}})|<\dfrac{1}{3}\widetilde{K}.$

Then integrating the equation for $\overline{U}$ in (4.48) from $B$ to $B_{1}$
yields

$\displaystyle
t_{b_{1}}^{-\frac{2\nu+1}{2}}\overline{U}(t_{b_{1}},\psi_{b_{1}})-t_{b}^{-\frac{2\nu+1}{2}}\overline{U}(t_{b},\psi_{b})=\displaystyle\int_{t_{b}}^{t_{b_{1}}}\bigg{\\{}\left(t^{2}\bar{f}_{2}-\dfrac{1}{2}\right)t^{-\frac{2\nu+3}{2}}\overline{V}+(t^{2}\bar{f}_{1}\overline{U}+t^{2}\bar{f}_{3})t^{-\frac{2\nu+3}{2}}\bigg{\\}}dt,$

which in view of (4.49) and (4.50) provides

$\displaystyle|\overline{U}(t_{b},\psi_{b})|$ $\displaystyle\leq
t_{b}^{\frac{2\nu+1}{2}}\bigg{\\{}t_{b_{1}}^{-\frac{2\nu+1}{2}}|\overline{U}(t_{b_{1}},\psi_{b_{1}})|+\displaystyle\int_{t_{b}}^{t_{b_{1}}}\left(\dfrac{9}{16}t^{-\frac{2\nu+3}{2}}|\overline{V}|+\dfrac{1}{16}t^{-\frac{2\nu+3}{2}}|\overline{U}|+\dfrac{1}{16}t^{-\frac{2\nu+3}{2}}\right)dt\bigg{\\}}$
$\displaystyle\leq
t_{b}^{\frac{2\nu+1}{2}}\bigg{\\{}\dfrac{1}{3}\widetilde{K}t_{b_{1}}^{-\frac{2\nu+1}{2}}+\displaystyle\int_{t_{b}}^{t_{b_{1}}}\left(\dfrac{9}{16}t^{-\frac{2\nu+3}{2}}\widetilde{K}+\dfrac{1}{16}t^{-\frac{2\nu+3}{2}}\widetilde{K}+\dfrac{1}{16}t^{-\frac{2\nu+3}{2}}\right)dt\bigg{\\}}$
$\displaystyle\leq
t_{b}^{\frac{2\nu+1}{2}}\bigg{\\{}\dfrac{1}{3}\widetilde{K}t_{b_{1}}^{-\frac{2\nu+1}{2}}+\widetilde{K}\displaystyle\int_{t_{b}}^{t_{b_{1}}}\dfrac{11}{16}t^{-\frac{2\nu+3}{2}}dt\bigg{\\}}$
$\displaystyle=t_{b}^{\frac{2\nu+1}{2}}\bigg{\\{}\dfrac{1}{3}\widetilde{K}t_{b_{1}}^{-\frac{2\nu+1}{2}}+\widetilde{K}\dfrac{11}{24}(t_{b}^{-\frac{2\nu+1}{2}}-t_{b_{1}}^{-\frac{2\nu+1}{2}})\bigg{\\}}<\dfrac{11}{24}\widetilde{K}<\widetilde{K},$

which contradicts the assumption that
$\overline{U}(t_{b},\psi_{b})=\widetilde{K}$. Therefore, the proof of the
Lemma is complete. ∎

By the arbitrariness of $A^{\prime}$ and $A^{\prime\prime}$ on
$\widehat{P^{\prime}D^{\prime}}$, we can acquire the uniform boundedness of
$\overline{U}$ and $\overline{V}$ up to the degenerate line
$\widehat{P^{\prime}D^{\prime}}$. Then we have the following Lemma:

######  Lemma 4.5.

The functions $\overline{L}$ and $L$ are uniformly bounded up to the
degenerate line $\widehat{P^{\prime}D^{\prime}}$.

###### Proof.

The proof of this Lemma directly follows from Lemma 4.1, 4.4 and (4.42).
Therefore, we omit the details. ∎

The uniform boundedness of $L$ leads to an important observation that $W=Z$ on
the degenerate line $\widehat{P^{\prime}D^{\prime}}$. Based on this property,
we can develop the uniform regularity of $W,Z$ and $L$ in the following Lemma:

######  Lemma 4.6.

The functions $W,Z$ and $L$ are uniformly $C^{\frac{1}{3}}$ while $r$ and
$\theta$ are uniformly $C^{1,\frac{1}{3}}$ continuous in the whole domain
$\Omega:P^{\prime}T^{\prime}D^{\prime}$, including the degenerate line
$\widehat{P^{\prime}D^{\prime}}$.

###### Proof.

We first prove the uniform continuity of $W$ at the degenerate line
$\widehat{P^{\prime}D^{\prime}}$. For any two points $(0,\psi_{1})$ and
$(0,\psi_{2})$ with $\psi_{1}<\psi_{2}$, we draw a negative characteristic
$l_{-}$ from $(0,\psi_{1})$ and a positive characteristic $l_{+}$ from
$(0,\psi_{2})$ and denote the intersection point of $l_{-}$ and $l_{+}$ by
$(t_{m},\psi_{m})$ such that the numbers $t_{m}$ and $\psi_{m}$ satisfy

$\displaystyle\psi_{m}$
$\displaystyle=\psi_{1}-\int_{0}^{t_{m}}\dfrac{ra\gamma_{a}t^{2}}{2\gamma
FW\sqrt{1-t^{2}}}dt=\psi_{2}+\int_{0}^{t_{m}}\dfrac{ra\gamma_{a}t^{2}}{2\gamma
FZ\sqrt{1-t^{2}}}dt.$ (4.52)

Now noting the expression of $F(t)$ and the fact that $0<a^{2}<1$,
$0<q<\hat{q}<1$, it is easy to see that there must exist two constants
$k_{1},k_{2}>0$ such that
$0<\dfrac{k_{1}}{2}\leq\dfrac{1}{F(t)}\leq\dfrac{k_{2}}{2}<\infty$. Therefore,
using Lemma 4.1, we must have

$\displaystyle\underline{M}t_{m}^{2}\leq\dfrac{ra\gamma_{a}t^{2}}{2\gamma
F\sqrt{1-t^{2}}}\leq\overline{M}t_{m}^{2}.$

for some positive constants $\underline{M}$ and $\overline{M}$. Then employing
(4.52) and using Lemma 4.1, we obtain

$\displaystyle\dfrac{2}{3}\underline{M}t_{m}^{3}\leq|\psi_{2}-\psi_{1}|\leq\dfrac{2}{3}\overline{M}t_{m}^{3}.$
(4.53)

Also, according to the uniform boundedness of $L$, we observe by (4.16) that
there must exists a constant $K>0$ such that $|\partial^{+}W|\leq K$ and
$|\partial^{-}Z|\leq K$. Therefore, utilizing (4.52), we eventually have

$\displaystyle|W(0,\psi_{2})-W(0,\psi_{1})|$
$\displaystyle=|Z(0,\psi_{2})-W(0,\psi_{1})|$
$\displaystyle\leq|Z(0,\psi_{2})-Z(t_{m},\psi_{m})|+|Z(t_{m},\psi_{m})-W(t_{m},\psi_{m})|$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ +|W(t_{m},\psi_{m})-W(0,\psi_{1})|$
$\displaystyle\leq Kt_{m}+2\underset{\Omega}{\max}|L|t_{m}+Kt_{m}$
$\displaystyle\leq(2K+2\underset{\Omega}{\max}|L|)t_{m}\leq\widehat{K}|\psi_{2}-\psi_{1}|^{\frac{1}{3}}$

for some positive constant $\widehat{K}$, which implies that the function
$\overline{W}$ is uniformly $C^{\frac{1}{3}}$ continuous on the degenerate
line $\widehat{P^{\prime}D^{\prime}}$.

From Lemma 4.3, it is easy to observe that the function $\overline{W}_{t}$ is
uniformly bounded in the region $\Omega$. Thus if $\psi_{1}=\psi_{2}$ then
using the boundedness of $\overline{W}_{t}$, we have

$\displaystyle|W(t_{1},\psi_{1})-W(t_{2},\psi_{2})|$
$\displaystyle=|W(t_{1},\psi_{1})-W(t_{2},\psi_{1})|$
$\displaystyle\leq\underset{\Omega}{\max}|W_{t}|.|t_{2}-t_{1}|$
$\displaystyle<K|(t_{1},\psi_{1})-(t_{2},\psi_{2})|^{\frac{1}{3}}.$

For the case $\psi_{1}<\psi_{2}$, there are two possible cases:

Case 1. If $\psi_{2}-\psi_{1}\leq t_{1}$ then we can choose $\nu=\frac{4}{3}$
in Lemma 4.4 and use mean value theorem to obtain

$\displaystyle|{W}(t_{2},\psi_{2})-{W}(t_{1},\psi_{1})|$
$\displaystyle\leq|{W}(t_{2},\psi_{2})-{W}(t_{1},\psi_{2})|+|{W}(t_{1},\psi_{2})-{W}(t_{1},\psi_{1})|$
$\displaystyle\leq K|t_{2}-t_{1}|+Kt_{1}^{-\frac{2}{3}}.|\psi_{2}-\psi_{1}|$
$\displaystyle\leq K|t_{2}-t_{1}|+K|\psi_{2}-\psi_{1}|^{\frac{1}{3}}$
$\displaystyle\leq K|(t_{2},\psi_{2})-(t_{1},\psi_{1})|^{\frac{1}{3}}$

for some uniform constant $K>0$.

Case 2. For the case $\psi_{2}-\psi_{1}>t_{1}$, we have

$\displaystyle|{W}(t_{2},\psi_{2})-{W}(t_{1},\psi_{1})|$
$\displaystyle\leq|{W}(t_{2},\psi_{2})-{W}(t_{1},\psi_{2})|+|{W}(t_{1},\psi_{2})-{W}(0,\psi_{2})|$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+|{W}(0,\psi_{2})-{W}(0,\psi_{1})|+|{W}(0,\psi_{1})-{W}(t_{1},\psi_{1})|$
$\displaystyle\leq|{W}_{t}|.|t_{2}-t_{1}|+|{W}_{t}|.t_{1}+\widehat{K}|\psi_{2}-\psi_{1}|^{\frac{1}{3}}+|{W}_{t}|.t_{1}$
$\displaystyle\leq
2K(|t_{2}-t_{1}|+|\psi_{2}-\psi_{1}|)+\widehat{K}|\psi_{2}-\psi_{1}|^{\frac{1}{3}}$
$\displaystyle\leq
K^{\prime}|(t_{2},\psi_{2})-(t_{1},\psi_{1})|^{\frac{1}{3}}$

for some uniform constant $K^{\prime}>0$, which implies that the function
${W}$ is uniformly $C^{\frac{1}{3}}$ continuous in the whole domain
$P^{\prime}T^{\prime}D^{\prime}$ including the degenerate line
$\widehat{P^{\prime}D^{\prime}}$. In a similar manner one can obtain the
uniform $C^{\frac{1}{3}}$ continuity of ${Z}$ and $L$ for any two points
$(t_{1},\psi_{1})$ and $(t_{2},\psi_{2})$ in the domain
$P^{\prime}T^{\prime}D^{\prime}$.

By using the same arguments as above, one can first show that the functions
$r(t,\psi)$ and $\theta(t,\psi)$ are uniformly $C^{\frac{1}{3}}$ continuous.
Furthermore, we recall from (4.12) that

$\displaystyle r_{t}$
$\displaystyle=-\dfrac{t\sqrt{1-t^{2}}r\cos\theta}{F(W+Z)},$ (4.54)
$\displaystyle r_{\psi}$
$\displaystyle=\dfrac{\gamma\sqrt{1-t^{2}}}{a\gamma_{a}(W+Z)}\left((W+Z)\sin\theta-2\sqrt{1-t^{2}}\cos\theta
L\right),$ (4.55)

which means that the function $r(t,\psi)$ is uniformly $C^{1,\frac{1}{3}}$
continuous in the whole domain $P^{\prime}T^{\prime}D^{\prime}$. The same
conclusion is also valid for the function $\theta(t,\psi)$ by (4.13). Hence,
the proof of the Lemma is finished. ∎

Finally, we draw a positive characteristic curve from the point
$D^{\prime}(0,\bar{\psi}(0))$ up to a point
$F^{\prime}(\bar{\delta},\tilde{\psi}(\bar{\delta}))$ lying on the boundary
$\widehat{P^{\prime}T^{\prime}}$ (see Figure 6) and combine the results of
subsections 4.3 and 4.4to achieve the following theorem:

######  Theorem 4.1.

Under the assumptions (4.9) and (4.15), the system (4.16) with boundary data
(4.8) and (4.14) possesses a global smooth solution
$\overline{W}(t,r),\overline{Z}(t,r)$ in the entire region
$P^{\prime}F^{\prime}D^{\prime}$ bounded by the curves
$\widehat{F^{\prime}D^{\prime}}$, $\widehat{P^{\prime}D^{\prime}}$ and
$\widehat{P^{\prime}F^{\prime}}$, where $F^{\prime}$ is the point
$(\bar{\delta},\psi(\bar{\delta}))$ and $\widehat{D^{\prime}F^{\prime}}$ is a
positive characteristic curve. Further, the solution
$(\overline{W},\overline{Z})(t,\psi)$ and the quantity $\overline{L}(t,\psi)$
are uniformly $C^{\frac{1}{3}}$ continuous up to the degenerate line
$\widehat{P^{\prime}D^{\prime}}$, i.e. $t=0$.

##  5 Solution in the physical plane

In this section, we recover a global smooth supersonic-sonic solution of the
system (4.16) in the physical plane using the solutions obtained in Theorem
4.1 in the partial hodograph plane via an inverse transformation.

###  5.1 Inversion

From Theorem 4.1, we know that the functions $(W,Z,r,\theta)(t,\psi)$ are
defined in the whole region $P^{\prime}F^{\prime}D^{\prime}$. Now we proceed
to construct the function $x(t,\psi)$ and then prove that the mapping
$(t,\psi)\longrightarrow(x,r)$ is a global one-to-one mapping.

Recalling the partial hodograph transformation (4.1), one can easily obtain

$\displaystyle
x_{t}=\dfrac{t\sqrt{1-t^{2}}r\sin\theta}{F(W+Z)},\leavevmode\nobreak\
\leavevmode\nobreak\
x_{\psi}=\dfrac{\sqrt{1-t^{2}}\gamma}{a\gamma_{a}(W+Z)}\left((W+Z)\cos\theta+2\sqrt{1-t^{2}}\sin\theta
W\right),$ (5.1)

or in other words

$\displaystyle\partial^{-}x=-\dfrac{tr(t\cos\theta-\sqrt{1-t^{2}}\sin\theta)}{2FW}.$
(5.2)

Then from any point $(\hat{t},\hat{\psi})$ in the region
$P^{\prime}F^{\prime}D^{\prime}$, we draw a negative characteristic curve
$\psi=\psi_{-}(t;\hat{t},\hat{\psi})\leavevmode\nobreak\ (t\geq\hat{t})$ up to
the boundary $\widehat{P^{\prime}F^{\prime}}$ at a unique point
$(\hat{t}^{\prime},\tilde{\psi}(\hat{t}^{\prime}))$ satisfying

$\displaystyle\begin{cases}\dfrac{d\psi_{-}(t;\hat{t},\hat{\psi})}{dt}=-\dfrac{r\gamma_{a}at^{2}}{2\gamma
F\sqrt{1-t^{2}}W}\left(t,\psi_{-}(t;\hat{t},\hat{\psi})\right),\\\
\psi_{-}(\hat{t};\hat{t},\hat{\psi})=\hat{\psi}(\hat{t}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\psi_{-}(\hat{t}^{\prime};\hat{t},\hat{\psi})=\hat{\psi}(\hat{t}^{\prime})\end{cases}$
(5.3)

Therefore, we integrate (5.2) along the negative characteristic from
$\hat{t}^{\prime}$ to $\hat{t}$ and use (5.3) to define the number
$x(\hat{t},\hat{\psi})$ as follows:

$\displaystyle
x(\hat{t},\hat{\psi})=\hat{x}(\tilde{\psi}(\hat{t}^{\prime}))+\displaystyle\int_{\hat{t}}^{\hat{t}^{\prime}}\dfrac{tr(t\cos\theta-\sqrt{1-t^{2}}\sin\theta)}{2FW}(t,\psi_{-}(t;\hat{t},\hat{\psi}))dt,$
(5.4)

where the function $\hat{x}(\psi)$ is defined in (4.7). Hence, by the
arbitrariness of $(\hat{t},\hat{\psi})$ one can conclude that the function
$x=x(t,\psi)$ can be defined in the whole region
$P^{\prime}F^{\prime}D^{\prime}$.

####  5.1.1 The mapping $(t,\psi)\longrightarrow(x,r)$ is globally injective

Noting the expressions of $r_{t},r_{\psi},x_{t}$ and $x_{\psi}$, it is
straightforward to see that
$j:=\dfrac{\partial(x,r)}{\partial(t,\psi)}=\dfrac{t(1-t^{2})r\gamma}{a\gamma_{a}F(t)(W+Z)}\neq
0,$ for $t>0$, which implies that the mapping $(t,\psi)\longrightarrow(x,r)$
is a local one to one mapping. In order to prove that the mapping is globally
one to one, including the line $t=0$, we only need to check the strict
monotonicity of $\phi$ along the level curves
$l^{\epsilon}:1-\varpi=\epsilon\geq 0$. We prove this by the method of
contradiction. Let us assume that there exist two distinct points
$(x_{1},r_{1})$ and $(x_{2},r_{2})$ in the region $PFD$ such that
$t_{1}=t_{2}$ and $\psi_{1}=\psi_{2}$ which implies that
$\cos\omega(x_{1},r_{1})=\cos\omega(x_{2},r_{2})$ and
$\psi(x_{1},r_{1})=\psi(x_{2},r_{2})$ such that both the points
$(x_{1},r_{1})$ and $(x_{2},r_{2})$ lies on the same level curve
$l^{\epsilon}:1-\sin\omega=\epsilon\geq 0$. Then we directly compute

$\displaystyle(\phi_{x},\phi_{r})\cdot(\varpi_{r},-\varpi_{x})=u\varpi_{r}-v\varpi_{x}=\dfrac{\gamma_{a}(t)\hat{F}_{1}(t)(W+Z)}{4ar\gamma\sqrt{1-t^{2}}}>0,$

using the fact that $W,Z>0$ by Lemma 4.1. Therefore, $\phi$ is monotonically
increasing function along each level curve of $l^{\epsilon}$ which contradicts
the assumption that $\phi(x_{1},r_{1})=\phi(x_{2},r_{2})$. Hence the mapping
is globally injective including the degenerate line $t=0$.

###  5.2 Solution of system (2.26)

We now construct the global smooth supersonic solution to system (2.26) using
the fact that the mapping $(t,\psi)\longrightarrow(x,r)$ is injective so that
we can obtain the functions $t=t(x,r)$ and $\psi=\psi(x,r)$ to define the
functions

$\displaystyle\theta=\theta(t(x,r),\psi(x,r)),\leavevmode\nobreak\
\leavevmode\nobreak\ \varpi=\sqrt{1-t^{2}(x,r)},\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ (x,r)\in PFD,$ (5.5)

where the region $PFD$ is bounded by the curves $\widehat{PF}$, $\widehat{PD}$
and $\widehat{DF}$ such that the curves $\widehat{PD}$ and $\widehat{DF}$ are
defined as follows

$\displaystyle\begin{cases}\widehat{PD}=\\{(x,r)|\varpi(x,r)=1,x\in[x_{1},x^{*}]\\},\\\
\widehat{DF}=\\{(x,r)|\psi(x,r)=\psi_{+}(t(x,r);\bar{\delta},\bar{\psi}(\bar{\delta})),x\in[x^{*},x^{**}]\\},\end{cases}$
(5.6)

where $x^{*}=x(0,\bar{\psi}(0))$ and the number $x^{**}$ satisfies
$\tilde{\psi}(x^{**},\varphi(x^{**}))=\psi_{+}(\tilde{t}(x^{**},\varphi(x^{**}));\bar{\delta},\tilde{\psi}(\bar{\delta}))$
such that the function $\psi_{+}$ is the solution of the ODE

$\displaystyle\begin{cases}\dfrac{d\psi_{+}(t;\bar{\delta},\tilde{\psi}(\bar{\delta}))}{dt}=\dfrac{r\gamma_{a}at^{2}}{2\gamma
F\sqrt{1-t^{2}}Z}(t,\psi_{+}(t;\bar{\delta},\tilde{\psi}(\bar{\delta}))),\leavevmode\nobreak\
\leavevmode\nobreak\ t\in[0,\bar{\delta}]\\\
\psi_{+}(\bar{\delta};\bar{\delta},\tilde{\psi}(\bar{\delta}))=\tilde{\psi}(\bar{\delta}).\end{cases}$
(5.7)

We can also get the coordinates of point $D$ and $F$ as
$(x^{*},r(0,\bar{\psi}(0)))$ and $(x^{**},\varphi(x^{**}))$, respectively. It
is easy to see that the functions $(\theta(x,r),\varpi(x,r))$ defined in (5.5)
satisfy the boundary condition (3.1) by the construction of
$(x(t,\psi),r(t,\psi))$. Now we proceed to verify that the function defined in
(5.5) satisfy the system (2.26) in $x-r$ plane.

####  5.2.1 Verification of solutions in $x-r$ plane

By performing a direct calculation, one can yield the following

$\displaystyle\begin{cases}\theta_{x}=\dfrac{t\sin\theta(W-Z)-\sqrt{1-t^{2}}\cos\theta(W+Z)+\sin^{2}\theta}{r},\vspace{0.2
cm}\\\
\theta_{r}=\dfrac{t\cos\theta(Z-W)-\sqrt{1-t^{2}}\sin\theta(W+Z)-\sin\theta\cos\theta}{r},\vspace{0.2
cm}\\\
\varpi_{x}=-\dfrac{\hat{F}_{1}(t)}{4a^{2}r}\left((W+Z)\sin\theta-2\sqrt{1-t^{2}}\cos\theta
L\right),\vspace{0.2 cm}\\\
\varpi_{r}=\dfrac{\hat{F}_{1}(t)}{4a^{2}r}\left((W+Z)\cos\theta+2\sqrt{1-t^{2}}\sin\theta
L\right).\end{cases}$ (5.8)

Therefore, using the definition of $\tilde{\partial}_{+}$ and (5.8), we obtain

$\displaystyle\tilde{\partial}_{+}\theta$
$\displaystyle=r(\cos\alpha\theta_{x}+\sin\alpha\theta_{r})$
$\displaystyle=\cos\alpha[\cos\omega\sin\theta(W-Z)-\sin\omega\cos\theta(W+Z)+\sin^{2}\theta]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\sin\alpha[\cos\omega\cos\theta(W-Z)-\sin\omega\sin\theta(W+Z)-\sin\theta\cos\theta]$
$\displaystyle=-\sin\omega\cos\omega(W-Z)-\cos\omega\sin\omega(W+Z)-\varpi\sin\theta$
$\displaystyle=-2\varpi\cos\omega W-\varpi\sin\theta.$

and

$\displaystyle\tilde{\partial}_{+}\varpi$
$\displaystyle=r(\cos\alpha\varpi_{x}+\sin\alpha\varpi_{r})$
$\displaystyle=\dfrac{F_{1}(\varpi)}{4a^{2}\gamma}\bigg{\\{}\cos\alpha[-\sin\theta(W+Z)+2\varpi\cos\theta
L]+\sin\alpha[\cos\theta(W+Z)+2\varpi\sin\theta L]\bigg{\\}}$
$\displaystyle=\dfrac{F_{1}(\varpi)}{4a^{2}\gamma}[\varpi(W+Z)+2\varpi\cos\omega
L]$
$\displaystyle=\dfrac{F_{1}(\varpi)}{4a^{2}\gamma}[\varpi(W+Z)+\varpi(W-Z)]$
$\displaystyle=2\varpi\dfrac{F_{1}(\varpi)}{4a^{2}\gamma}W$

Therefore, we have

$\displaystyle\tilde{\partial}_{+}\theta+\dfrac{4a^{2}\gamma\cos\omega}{F_{1}(\omega)}\tilde{\partial}_{+}\varpi$
$\displaystyle=[-2\varpi\cos\omega
W-\varpi\sin\theta]+\dfrac{4a^{2}\gamma\cos\omega}{F_{1}(\omega)}\left(2\omega\dfrac{F_{1}(\omega)}{4a^{2}\gamma}W\right)$
$\displaystyle=-\varpi\sin\theta,$

Hence, the functions $\theta(x,r)$ and $\varpi(x,r)$ satisfy first equation of
(2.26). In a similar manner one can prove that $\theta(x,r)$ and $\varpi(x,r)$
satisfy second equation of $\eqref{eq: 2.21}$ and therefore, $\theta(x,r)$ and
$\varpi(x,r)$ is the solution of system (2.26) with boundary conditions (3.1).

###  5.3 Regularity of angle variables and sonic boundary in the physical
plane

We now discuss the regularity of $\theta(x,r)$ and $\varpi(x,r)$,
respectively. Using (5.8), Lemma 4.1 and Lemma 4.5, it is easy to see that the
functions $\theta_{x},\theta_{r},\varpi_{x}$ and $\varpi_{r}$ are uniformly
bounded, implying $\theta(x,r)$ and $\varpi(x,r)$ are uniformly Lipschitz
continuous. We can actually prove that the functions
$\theta_{x},\theta_{r},\varpi_{x}$ and $\varpi_{r}$ are uniformly
$C^{\frac{1}{6}}$ continuous. We prove this result in the following Lemma.

######  Lemma 5.1.

Let $f(t,r)$ be a $C^{\frac{1}{3}}$ function defined on the whole region
$P^{\prime}F^{\prime}D^{\prime}$. Then if we denote
$\bar{f}(x,r)=f(t(x,r),\psi(x,r))$ then the function $\bar{f}(x,y)$ is
uniformly $C^{\frac{1}{6}}$ continuous in the whole region $PFD$.

###### Proof.

Let $(x^{\prime},r^{\prime})$ and $(x^{\prime\prime}.r^{\prime\prime})$ be any
two points in $PFD$ and let $(t^{\prime},\psi^{\prime})$ and
$(t^{\prime\prime},\psi^{\prime\prime})$ be the images of
$(x^{\prime},r^{\prime})$ and $(x^{\prime\prime},r^{\prime\prime})$ in the
region $P^{\prime}F^{\prime}D^{\prime}$. Then we evaluate

$\displaystyle|\bar{f}(x^{\prime\prime},r^{\prime\prime})-\bar{f}(x^{\prime},r^{\prime})|$
$\displaystyle=|f(t^{\prime\prime},\psi^{\prime\prime})-f(t^{\prime},\psi^{\prime})|$
$\displaystyle\leq
K|(t^{\prime\prime},\psi^{\prime\prime})-(t^{\prime},\psi^{\prime})|^{\frac{1}{3}}=K\left(|t^{\prime\prime}-t^{\prime}|^{2}+|\psi^{\prime\prime}-\psi^{\prime}|^{2}\right)^{\frac{1}{6}}$

for some uniform constant $K>0$. Now for the term
$|\psi^{\prime\prime}-\psi^{\prime}|$, one has

$\displaystyle|\psi^{\prime\prime}-\psi^{\prime}|=|\phi(x^{\prime\prime},r^{\prime\prime})-\phi(x^{\prime},r^{\prime})|\leq
K|(x^{\prime\prime},r^{\prime\prime})-(x^{\prime},r^{\prime})|,$

by the uniform Lipschitz continuity of $\phi(x,r)$.

Again, for the term $|t^{\prime\prime}-t^{\prime}|^{2}$, we have

$\displaystyle|t^{\prime\prime}-t^{\prime}|^{2}$
$\displaystyle\leq|t^{\prime\prime}-t^{\prime}|.|t^{\prime\prime}+t^{\prime}|=|t^{\prime\prime
2}-t^{\prime 2}|$
$\displaystyle\leq|(1-\varpi^{2}(x^{\prime\prime},r^{\prime\prime}))-(1-\varpi^{2}(x^{\prime},r^{\prime}))|$
$\displaystyle\leq|\varpi^{2}(x^{\prime\prime},r^{\prime\prime})-\varpi^{2}(x^{\prime},r^{\prime})|\leq
2|\varpi(x^{\prime\prime},r^{\prime\prime})-\varpi(x^{\prime},r^{\prime})|$
$\displaystyle\leq
K|(x^{\prime\prime},r^{\prime\prime})-(x^{\prime},r^{\prime})|$

in view of the uniform Lipschitz continuity of $\varpi$. Hence, we directly
compute

$\displaystyle|\bar{f}(x^{\prime\prime},r^{\prime\prime})-\bar{f}(x^{\prime},r^{\prime})|\leq
K|(x^{\prime\prime},r^{\prime\prime})-(x^{\prime},r^{\prime})|^{\frac{1}{6}},$
(5.9)

which implies that the function $\bar{f}$ is uniformly $C^{\frac{1}{6}}$
continuous. Therefore, Lemma is proved. ∎

Hence, by using the Lemma 4.5 and Lemma 5.1, it is easy to see that the
functions $(W,Z,L,\theta)(t(x,r),\psi(x,r))$ are uniformly $C^{\frac{1}{6}}$
continuous in the whole region $PFD$ up to the sonic boundary $\widehat{PD}$.
Therefore, using (5.8) we observe that the functions $\varpi(x,r)$ and
$\theta(x,r)$ are uniformly $C^{1,\frac{1}{6}}$ continuous in the domain $PFD$
up to the sonic curve $\widehat{PD}$.

Moreover, we directly apply (5.8), Lemma 4.1 and Lemma 4.5 to observe that

$\displaystyle
0<\widetilde{k}\leq(\varpi_{x})^{2}+(\varpi_{r})^{2}=\dfrac{(F_{1}(\varpi))^{2}}{16a^{4}r^{2}}\left((W+Z)^{2}+4\varpi^{2}L^{2}\right)\leq\widetilde{K}<\infty,$

for some positive constants $\widetilde{K}$ and $\widetilde{K}$. This fact
implies that the level curve $\varpi(x,r)=\epsilon\geq 0$ is $C^{1}$
continuous. Furthermore, due to Lemmas 4.5 and 5.1, the level curve
$\varpi(x,r)$ and eventually the sonic curve $\widehat{PD}$ are actually
$C^{1,\frac{1}{6}}$ continuous.

###  5.4 $\widehat{DF}$ is a negative characteristic curve

Since $\widehat{D^{\prime}F^{\prime}}$ is a positive characteristic curve so
in order to prove that $\widehat{DF}$ is a negative characteristic curve, it
is enough to prove that the mapping $(t,\psi)\longrightarrow(x,r)$ transforms
a positive characteristic curve in $t-\psi$ plane into a negative
characteristic curve in $x-r$ plane. We prove this in the following Lemma.

######  Lemma 5.2.

A positive characteristic curve in $(t,\psi)$ plane is transformed into a
negative characteristic curve in $(x,r)$ plane under the transformation (5.5).

###### Proof.

To prove it, we differentiate the equality $\psi(x,r)=\psi_{+}(t(x,r))$ with
respect to $x$ and use the fact $\psi_{+}^{\prime}(t)=\lambda$ to get

$\displaystyle\dfrac{dr}{dx}=-\dfrac{\psi_{x}-\lambda t_{x}}{\psi_{r}-\lambda
t_{r}}=-\dfrac{\lambda\sqrt{1-t^{2}}\omega_{x}-\phi_{x}}{\lambda\sqrt{1-t^{2}}\omega_{r}-\phi_{r}}.$
(5.10)

Then by exploiting (5.8) in (5.10) and applying (5.5) yields

$\displaystyle\dfrac{dr}{dx}$
$\displaystyle=\dfrac{-2\sqrt{1-t^{2}}Z\cos\theta+t(W+Z)\sin\theta-2\sqrt{1-t^{2}}\cos\theta
Lt}{2\sqrt{1-t^{2}}Z\sin\theta+t(W+Z)\cos\theta+2\sqrt{1-t^{2}}\sin\theta Lt}$
$\displaystyle=\dfrac{\cos\omega\sin\theta-\sin\omega\cos\theta}{\cos\omega\cos\theta+\sin\omega\sin\theta}=\dfrac{\sin\beta}{\cos\beta}=\Lambda_{-},$

which implies that the curves defined by the equality
$\psi(x,r)=\psi_{+}(t(x,r))$ are negative characteristics in the $x-r$ plane.
∎

Now since $\dfrac{da}{dq}<0$ and $\dfrac{d\gamma}{dq}>0$, then we must have
$\dfrac{da}{d\gamma}<0$ or in other words $\gamma:=\gamma(a)$. Then we combine
$a=a(\varpi)$ and (5.5) together with
$u=\dfrac{a\gamma_{a}\cos\theta}{\gamma\sin\omega},v=\dfrac{a\gamma_{a}\sin\theta}{\gamma\sin\omega}$
to define the functions $(a,u,v)(x,r)$ such that

$\displaystyle a(x,r)=a(\varpi(x,r)),\leavevmode\nobreak\
u=\dfrac{a(\varpi(x,r))\gamma_{a(\varpi(x,r))}\cos\theta(x,r)}{\gamma(a(\varpi(x,r)))\varpi(x,r)},\leavevmode\nobreak\
v=\dfrac{a(\varpi(x,r))\gamma_{a(\varpi(x,r))}\sin\theta(x,r)}{\gamma(a(\varpi(x,r)))\varpi(x,r)}$
(5.11)

is the classical solution of (2.11).

In view of assumptions (3.3), (3.4), Theorem 4.1, Lemma 5.1 and Lemma 5.2, we
have the following result:

######  Theorem 5.1.

Let $\widehat{PE}:r=\varphi(x)$ is an increasing and concave smooth streamline
of a 3-D steady axisymmetric isentropic irrotational relativistic flow such
that the Mach number $M$ increases along $\widehat{PE}$ with $M=1$ at the
point $P$ and $\varphi^{\prime}$ and $\varpi$ satisfy (3.3) and (3.4). Then
there exists a smooth sonic curve $\widehat{PD}$ and a negative characteristic
curve $\widehat{DF}$ such that the boundary value problem (2.26)-(3.1) has a
smooth supersonic solution $(\theta,\varpi)$ in the region $PFD$, where $F$ is
a point lying on the streamline $\widehat{PE}$. Furthermore, solution
$(\theta,\varpi)(x,r)$ is uniformly $C^{1,\frac{1}{6}}$ continuous in the
whole region $PFD$ while the sonic curve $\widehat{PD}$ is $C^{1,\frac{1}{6}}$
continuous.

##  6 Conclusions

In this article, we considered three-dimensional axisymmetric steady
isentropic relativistic Euler equations with a general convex pressure and
proved the global existence and regularity of solution of a supersonic-sonic
patch arising in the modified Frankl problem. Using the characteristic
decompositions of angle variables and a partial hodograph transformation, we
were able to prove that solution is uniformly $C^{1,\frac{1}{6}}$ continuous.
Moreover, we proved that the sonic boundary is $C^{1,\frac{1}{6}}$ continuous.
The study of such supersonic-sonic patch problems is quite crucial in the
context of transonic flows. Here we constructed solution up to a negative
characteristic curve $\widehat{DF}$. However, in the future, we will try to
construct a global smooth supersonic solution of the modified Frankl problem
for relativistic Euler equations with arbitrary equation of state up to the
positive characteristic curve $\widehat{EO}$ by solving a free boundary value
problem and using the symmetry of the airfoil.

## Acknowledgments

The first author (RB) gratefully acknowledges the research support from the
University Grant Commission, Government of India. The second author (TRS)
would like to thank SERB, DST, India (Ref. No. MTR/2019/001210) for its
financial support through the MATRICS grant.

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