\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 248, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/248\hfil Interacting rarefaction waves]
{Interacting rarefaction waves for the unsteady transonic small disturbance
equation}

\author[I. Jegdi\'c, K. Jegdi\'c \hfil EJDE-2016/248\hfilneg]
{Ilija Jegdi\'c, Katarina Jegdi\'c}

\address{Ilija Jegdi\'c \newline
Department of Mathematics,
Texas Southern University,
3100 Cleburne Street, 
Houston, TX 77004, USA}
\email{ilija.jegdic@tsu.edu}

\address{Katarina Jegdi\'c \newline
Department of Mathematics and Statistics,
University of Houston - Downtown,
1 Main St, Houston, TX 77002, USA}
\email{jegdick@uhd.edu}


\thanks{Submitted November 16, 2015. Published September 14, 2016.}
\subjclass[2010]{35L65}
\keywords{Unsteady transonic small disturbance; mixed type system; 
\hfill\break\indent rarefaction waves; shocks}

\begin{abstract}
 We consider a Riemann problem for the unsteady transonic small disturbance 
 equation  that results in two rarefaction waves. We write the problem in 
 self-similar  and parabolic coordinates and we obtain a system that changes 
 type from hyperbolic to elliptic. We use the characteristic decomposition 
 equations to study the complicated interaction of these two rarefaction waves
 in the hyperbolic region. We obtain local existence of the solution and
 we derive various properties of the solution and of the characteristic curves.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\newcommand{\R}{\mathbb{R}}
\newcommand{\abs}[1]{\lvert#1\rvert}


\section{Introduction and summary of this article}

To better understand the structure of the solutions to the Cauchy problem for
the two-dimensional Euler gas dynamics equations, scientists have considered 
simpler models  with sectorially constant initial data known as Riemann initial 
data. There has been a lot of progress in the past several
decades in this research area using numerical and analytical approaches. 
Some of the simplified models that have been considered are the isentropic 
gas dynamics equations, the pressure-gradient system, the nonlinear
wave system, the potential flow and the unsteady transonic small 
disturbance (UTSD) equation (see Keyfitz \cite{K}).

Conjectures on the structure of the solutions to isentropic and full gas 
dynamics equations, where the Riemann initial data is constant in four quadrants 
and each discontinuity results in a shock, rarefaction
wave or a slip line, are given by Zhang and Zheng \cite{ZZ}, 
revealing very complicated interactions using the theory
of generalized characteristics. The numerical simulations are given 
by Schulz-Rinne et al.\ \cite{S}.
For the analytical and numerical studies of solutions to the pressure-gradient 
system, with Riemann data posed in four quadrants, see \cite{ZLZ, Zbook}.

A Riemann problem with data constant in four quadrants that results in 
interacting rarefaction waves is considered numerically for the full gas 
dynamics equations by Glimm et al.\ in \cite{Glimm}. They
observed reflection of characteristics from the sonic curves and formation of shocks. 
These regions where one family of characteristics starts on a sonic curve and 
ends on a transonic shock are known in the literature as semi-hyperbolic patches 
and have been studied using the characteristic decomposition equations. 
For the study of semi-hyperbolic patches for the pressure-gradient
system, see Song and Zheng \cite{SZ}, for the Euler equations, 
see Li and Zheng \cite{LZ}, for
the nonlinear wave system, see \cite{HW} by Hu and Wang, and for the UTSD equation, 
see Jegdi\'c and Jegdi\'c \cite{IK2} and  Tesdall and Keyfitz \cite{TK}.
Conjecture on the structure of the solution for the nonlinear wave
system with diverging rarefaction waves was given  by Jegdi\'c and 
Jegdi\'c \cite{IK1}. A Riemann
problem for the pressure-gradient system with data constant in four quadrants 
resulting in interacting rarefaction waves without the reflection of waves 
from the sonic curves was studied by Bang \cite{B}.

Shock reflection phenomena for the UTSD
equation have been studied in \cite{BH, CKK1, CKK2, J2}. 
In this article we are interested in interaction of rarefaction waves and 
in \S 2 we formulate the initial value problem
we consider. As in \cite{CKK1, CKK2, IK2, J2}, we write the problem
in self-similar coordinates and obtain a reduced system that changes type 
from hyperbolic to elliptic with boundary data given in the far field. 
Using the generalized characteristics as in \cite{ZLZ, ZZ, Zbook},  we derive
conjectures on two configurations of the solution. We notice that the 
hyperbolic region consists of fully-hyperbolic, semi-hyperbolic and, 
in the case of the second configuration,
impracticably hyperbolic subregions
(see  Zheng \cite{Zbook} for this terminology for the pressure-gradient
system). In \S 3 we recall the characteristic decomposition equations
from \cite{IK2}, which we use to study solution locally in the fully-hyperbolic 
and semi-hyperbolic subdomains. We prove local existence of smooth solution 
in these subdomains and we derive statements on the monotonicity
of solution along characteristics and various properties of characteristics curves.


\section{Formulation of the problem and structure of the solution}

In this section we formulate the initial value problem we consider and 
we rewrite it using self-similar
coordinates. We use the generalized characteristics to
pose conjectures on the structure of two possible solutions.

\subsection{Formulation of the initial value problem}

We consider the unsteady transonic small disturbance equation
\begin{equation}\label{utsd}
\begin{gathered}
u_t + uu_x + v_y  =  0, \\
-v_x+u_y  =  0, \\
\end{gathered}
\end{equation}
where $(x,y) \in \R^2$, $t \in [0,\infty)$, and
 $(u,v): [0, \infty) \times \R^2 \to \R$ are
velocities in $x$- and $y$-directions, respectively. We use the notation $U=(u,v)$.
Given a parameter $a>0$, we consider initial data consisting of three 
constant states posed in three sectors (see Figure \ref{fig1})
\begin{equation} \label{initial}
U(0,x,y)=\begin{cases}
U_2=(-1,a), & y > \max\{x/a, 0\}, \\
U_3=(-1,-a), & y< \min \{-x/a,0\}, \\
U_1=(0,0), & \text{otherwise}.
\end{cases}
\end{equation}

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm} \scriptsize
\begin{picture}(35,35)(0,0)
\put(0,15){\vector(1,0){35}}
\put(15,0){\vector(0,1){35}}
\put(15,15){\line(1,1){15}}
\put(15,15){\line(1,-1){15}}
\put(12,34){$y$}
\put(26,24){$x=ay$}
\put(8,22){$U_2$}
\put(26,19){$U_1$}
\put(33,12){$x$}
\put(8,8){$U_3$}
\put(26,6){$x=-ay$}
\end{picture}
\end{center} 
\caption{Initial data} \label{fig1}
\end{figure}

\subsection{Formulation of the problem in self-similar coordinates}

We write system \eqref{utsd} in self-similar coordinates $\xi=x/t$ and $\eta=y/t$
\begin{equation}\label{mixed}
\begin{gathered}
(u-\xi) u_{\xi} - \eta u_{\eta} + v_{\eta}  =  0, \\
-v_{\xi} + u_{\eta}  =  0.
\end{gathered}
\end{equation}
The eigenvalues of this system are
\begin{equation}
\label{ev_xieta}
\Lambda_{\pm} = \frac{\eta \pm \sqrt{\eta^2 + 4(\xi - u)}}{2(\xi-u)},
\end{equation}
implying that \eqref{mixed} changes type across the sonic parabola
\[
P_u : \xi + \frac{\eta^2}{4}=u,
\]
and \eqref{mixed} is hyperbolic outside of $P_u$ and elliptic inside $P_u$.
Hence, under this change of variables, the initial data \eqref{initial} 
becomes boundary data given on the ``parabola in the far field''
\[
U(\xi,\eta)=\left\{
\begin{array}{ll}
U_2, & \eta > \max\{\xi/a, 0\}, \, \xi + \eta^2/4 \to \infty\\
U_3, & \eta < \min \{-\xi/a,0\}, \, \xi + \eta^2/4 \to \infty\\
U_1, & \xi>0,\, -\xi/a < \eta < \xi/a, \, \xi + \eta^2/4 \to \infty.
\end{array}
\right.
\]

Using the theory of one-dimensional hyperbolic conservation laws,
we obtain that each of the
discontinuities in the far field resolves in a rarefaction wave.
 More precisely, we obtain a rarefaction wave
\[
R_{12}
\left\{
\begin{array}{l}
U=U(\xi-a\eta)\\
u=\xi-a\eta-a^2 \\
v=-a u
\end{array}
\right.
\]
connecting the states $U_1$ on the left and $U_2$ on the right, 
and a rarefaction wave
\[
R_{13}
\left\{
\begin{array}{l}
U=U(\xi+a\eta)\\
u=\xi+a\eta-a^2 \\
v=a u
\end{array}
\right.
\]
connecting the states $U_3$ on the left and $U_1$ on the right. 
Therefore, the left and the right borders
of $R_{12}$ are given by $\xi=a\eta+a^2$ and $\xi=a\eta+a^2-1$, 
respectively, and the left and the right
borders of $R_{13}$ are given by $\xi=-a\eta +a^2-1$ and 
$\xi=-a\eta +a^2$, respectively (see Figure \ref{fig2}).

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm} \scriptsize
\begin{picture}(50,50)(0,0)
\put(0,25){\vector(1,0){50}}
\put(25,0){\vector(0,1){50}}
\qbezier(10,10)(40,25)(10,40)
\qbezier(0,10)(30,25)(0,40)
\put(40,40){\line(1,1){8}}
\put(39,41){\line(1,1){8}}
\put(41,39){\line(1,1){8}}
\put(42,38){\line(1,1){8}}
\put(40,10){\line(1,-1){8}}
\put(39,9){\line(1,-1){8}}
\put(41,11){\line(1,-1){8}}
\put(42,12){\line(1,-1){8}}
\put(22,48){$\eta$}
\put(13,40){$P_0$}
\put(0,35){$P_{-1}$}
\put(48,22){$\xi$}
\put(49,49){$R_{12}$}
\put(49,0){$R_{13}$}
\put(35,48){$U_2$}
\put(35,2){$U_3$}
\put(47,35){$U_1$}
\end{picture}
\end{center}
\caption{Solution in the far field and the sonic parabolas $P_0$ and $P_{-1}$}
\label{fig2}
\end{figure}

\subsection{Two possible structures of the solution}

We study the interaction of rarefaction waves $R_{12}$ and $R_{13}$ 
coming from the far field with
the sonic parabola $P_0: \xi+ \eta^2/4 =0$ for the state $U_1$ and the
 sonic parabola
$P_{-1}: \xi+\eta^2/4=-1$ for the states $U_2$ and $U_3$.

The rarefaction waves $R_{12}$ and $R_{13}$
start to interact at the point $A(a^2,1)$. The problem is symmetric about 
the $\xi$-axis and we focus
our attention to the upper half-plane. We find the $\Lambda_{\pm}$-characteristics 
at $A$ by integrating
\[
\frac{d\eta}{d\xi} = \Lambda_{\pm}
\]
and using that
$\xi-u=a\eta+a^2$ across $R_{12}$. We compute that
$\Lambda_{+}$ characteristics at $A$ is
\[
\eta = \frac{\xi}{a} - a,
\]
which is the left border of $R_{12}$,
and that
$\Lambda_-$ characteristics at $A$ is
\[
\frac{(a+\eta)^2}{2}=-\xi + \frac{3a^2}{2},
\]
which penetrates $R_{12}$ and intersects the right border of $R_{12}$ at the point
\[
B(-a^2 + a \sqrt{4a^2+2}-1, -2a + \sqrt{4a^2+2})
\]
(see Figure \ref{fig3}).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3}
\\
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)(0,-8)
\put(-41,57){$C$}
\put(-14,57){$P_0$}
\put(-2,57){$y$}
\put(12,52){$U_2$}
\put(39,52){$R_{12}$}
\put(-31,41){$F$}
\put(20,40){$B$}
\put(-19,38){$D$}
\put(-17,33){$E$}
\put(-9,31){$U_4$}
\put(38,31){$U_1$}
\put(-31,25){$P_{-1}$}
\put(9,25){$M$}
\put(35,25){$A$}
\put(42,25){$\xi$}
\put(-17,22){$E_1$}
\put(19,15){$B_1$}
\put(-31,12){$F_1$}
\put(-20,12){$D_1$}
\put(12,3){$U_3$}
\put(39,3){$R_{13}$}
\put(-41,-2){$C_1$}
\end{picture}
\end{center}
\caption{Configuration I}
\label{fig3}
\end{figure}
It is easy to check that the point $B$ is outside of the parabola $P_{-1}$, 
implying that the point $B$ is hyperbolic with respect to the state $U_2$. 
The straight $\Lambda_+$ characteristics of the rarefaction
wave $R_{12}$ curve bellow the parabolic arc $\widehat{AB}$.
Since the problem is symmetric with respect to
$\xi$-axis, after meeting at the point $A$, we conclude that the rarefaction
 waves $R_{12}$ and $R_{13}$
interact to form the penetration region $ABMB_1A$. The problem in this region 
is a Goursat type problem with data given on characteristic boundaries 
$\widehat{AB}$ and $\widehat{AB_1}$.

Upon exiting the boundaries $\widehat{BM}$ and $\widehat{B_1M}$,
the curved characteristics become again straight forming two simple waves. 
We again focus on the upper half-plane and we find the $\Lambda_-$ characteristics 
at $B$ which is a line tangent to the sonic parabola $P_{-1}$ at the point
\begin{equation} \label{C}
C(-5a^2+2a\sqrt{4a^2+2}-3, -2a + 2 \sqrt{4a^2+2}).
\end{equation}
It is easy to show that the characteristics $\widehat{ABC}$ has continuous 
first derivatives at $A$ and $B$.

Next, we follow $\Lambda_+$ characteristics at $C$. This characteristics 
goes backward through the simple wave in the upper half-plane and exists 
the simple wave at the point $D$. Afterwords, this characteristics
becomes straight ending tangentially at the point $E$
on the sonic parabola for the constant state $U_4$ which is a solution to the
Goursat problem in the region $MDEE_1D_1M$ with data given on characteristics 
boundaries $\overline{MD}$ and $\overline{MD_1}$.
The straight $\Lambda_-$ characteristics of the simple wave in the upper
half-plane become curved to the right of $\widehat{CD}$ until they become
 sonic along $\widehat{CF}$. We can think of these
$\Lambda_-$ characteristics reflecting of the sonic boundary $\widehat{FC}$
as $\Lambda_+$ characteristics.
After passing through $\widehat{DF}$, these $\Lambda_+$
characteristics become again straight until they reach the sonic boundary 
$\widehat{FE}$.

Therefore, the boundary of the interaction domain consists of the 
$\Lambda_-$ characteristic curve $\widehat{ABC}$,
$\Lambda_+$ characteristic curve $\widehat{AB_1C_1}$
and the part of the sonic parabola $P_{-1}$ outside of the arc
$\widehat{CC_1}$.
The exterior of this domain consists of two rarefaction waves and three constant 
states. The interaction domain consists of the elliptic and hyperbolic subdomains.

The problem in the unbounded region with boundary
\[
\widehat{CFEE_1C_1} \cup \{(\xi, \eta) \in P_{-1}:
\eta > \eta_C \text{ or } \eta < \eta_{C_1} \}
\]
is {\it elliptic}, while the problem in the bounded region $ABCFEE_1F_1C_1B_1A$ is
{\it hyperbolic}. Moreover,
the problem in the region $ABCDEE_1D_1C_1B_1A$ is
{\it fully-hyperbolic}
since both sets of characteristics can be traced back to the parabola in the 
far-field. The regions $CDEFC$ and $C_1D_1E_1F_1C_1$ are {\it partially hyperbolic} 
and in the literature known as
{\it semi-hyperbolic patches} as one set of characteristics can be traced to 
the far-field parabola and the
other set of characteristics starts and ends on sonic curves.
 More on this terminology for the pressure-gradient system could be found 
in Zheng  \cite{Zbook}.
In particular, the semi-hyperbolic patches for the unsteady transonic small 
disturbance equation were studied in \cite{IK2}
where it is shown that the characteristics which start on a sonic curve form 
an envelope that is overtaken by a transonic shock. Hence, we conjecture that 
curved $\Lambda_+$ characteristics exiting the
boundary $\widehat{DF}$ as straight characteristics end on a transonic shock.
 Numerical examples of interacting rarefaction waves for the full Euler gas 
dynamics equations are given by Glimm et al.\ \cite{Glimm} 
showing a complicated structure involving semi-hyperbolic patches and
reflection of characteristics which start on a sonic curve and end on a 
transonic shock. Tesdall and Keyfitz \cite{TK} study the interaction of 
the rarefaction wave with the sonic boundary and show numerically that
the the resulting shock starts (or ends) in the supersonic region.

The second possible structure of the solution is illustrated in Figure \ref{fig4}. 
In this case, the $\Lambda_+$ characteristics $DE$ and
the $\Lambda_-$ characteristics $D_1E_1$ intersect each other before becoming 
sonic. Let $X$ denote their intersection. The region
$EXE_1E$ is known in the literature (see \cite{Zbook}) as {\it impracticably 
hyperbolic} since at each point in this region neither set
of characteristics could be traced back to the far field parabola.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4}
\\
\setlength{\unitlength}{1mm}
\begin{picture}(0,0)(0,-8)
\put(-41,57){$C$}
\put(-14,57){$P_0$}
\put(-2,57){$y$}
\put(12,52){$U_2$}
\put(39,52){$R_{12}$}
\put(-31,41){$F$}
\put(20,40){$B$}
\put(-11.5,33.5){$D$}
\put(-22,33){$E_1$}
\put(-8,31){$U_4$}
\put(38,31){$U_1$}
\put(-31,25){$P_{-1}$}
\put(-13,25){$X$}
\put(9,25){$M$}
\put(35,25){$A$}
\put(42,25){$\xi$}
\put(-22,21){$E$}
\put(-11,21){$D_1$}
\put(19,15){$B_1$}
\put(-31,12){$F_1$}
\put(12,3){$U_3$}
\put(39,3){$R_{13}$}
\put(-41,-2){$C_1$}
\end{picture}
\end{center}
\caption{Configuration II}
\label{fig4}
\end{figure}

\section{Analysis in the hyperbolic part of the interaction region}

The rest of the paper is about the study of the solution structure in 
Configuration I (Figure \ref{fig3})
and we focus our attention to the hyperbolic part of the interaction region.
 We study the solution in three subdomains: penetration region $ABMB_1A$
 between $R_{12}$ and $R_{13}$, simple domain region
$BCDMB$ and the semi-hyperbolic patch $CDEFC$. In \S 3.1 we write the problem 
in parabolic coordinates and we recall the characteristic decomposition equations 
derived in \cite{IK2}. We use these equations to study Goursat problems in
domains $ABMB_1A$ and $BCDMB$, where the data is known
along two characteristic boundaries.

In the semi-hyperbolic patch $CDEFC$, positive characteristics reflect of 
the sonic curve $\widehat{FC}$ and the solution in $CDFC$ depends on the 
solution in the elliptic region with $\widehat{FC}$
being a free sonic boundary. To the best of our knowledge, understanding of 
this dependence is still an open problem and our future work consists
of computing detailed numerical simulations using the numerical method from  
\cite{I} by Jegdi\'c to better understand this dependence.
As in earlier studies of semi-hyperbolic patches \cite{HW, IK2, LZ, SZ},
to study the solution in the region $CDEFC$, we
assume that the location of the characteristic curve $\widehat{DF}$ is known.

In \S 3.2 and 3.4 we prove existence of  local smooth solutions in regions 
$ABMB_1A$ and $CDFC$, and we obtain various properties of global smooth 
solutions in these regions. Our future work consists
of using these properties and ideas from \cite{Dai} by Dai
to extend these local solutions to the entire regions $ABMB_1A$ and
$CDFC$.


\subsection{Formulation of the problem in parabolic coordinates and 
characteristic decomposition}

We consider system \eqref{mixed}
in parabolic coordinates $(\rho, \eta)$ where $\rho = \xi + \eta^2/4$
\begin{equation}\label{rhoeta}
\begin{gathered}
(u-\rho)u_{\rho}- \frac{\eta}{2}u_{\eta} + v_{\eta}  =   0, \\
\frac{\eta}{2}u_{\rho} - v_{\rho} + u_{\eta} =  0,
\end{gathered}
\end{equation}
with eigenvalues
\[
\lambda_{\pm} = \pm \frac{1}{\sqrt{\rho-u}}.
\]
From \cite{IK2} we recall the definitions of the
directional derivatives along $\Lambda_{\pm}$ and $\lambda_{\pm}$
characteristic curves
\begin{equation}\label{dir_der}
\partial^{\pm} = \partial_{\xi} + \Lambda_{\pm} \partial_{\eta}
\quad \text{and} \quad
\partial_{\pm} = \partial_{\eta} + \lambda_{\pm}^{-1} \partial_{\rho}
 = \partial_{\eta} \pm \sqrt{\rho-u} \partial_{\rho},
\end{equation}
the following relationships
\begin{equation}
\label{pmL}
\partial^{\pm} \Lambda_{\pm} = \frac{\Lambda_{\pm}^3}{\eta \Lambda_{\pm} +2} 
\partial^{\pm} u
\quad \text{and} \quad
\partial_{\pm} u = \frac{\eta \Lambda_{\pm}+2}{\Lambda_{\pm}^4}\partial^{\pm} \Lambda_{\pm},
\end{equation}
and the characteristic decomposition equations
\begin{equation}
\label{decomposition}
\partial_{\pm} \partial_{\mp} u= Q(\partial_{\pm} u 
- \partial_{\mp} u) \partial_{\mp}u,
\end{equation}
where
\begin{equation}
\label{Q}
Q = \frac{1}{4(\rho-u)}=-\frac{\lambda_+ \lambda_-}{4}.
\end{equation}
Notice that equations \eqref{decomposition} could be rewritten as
\begin{equation}\label{dec_new}
\partial_{\pm} \big(\frac{1}{\partial_{\mp} u} \big)
+ Q \partial_{\pm}u \frac{1}{\partial_{\mp}u} =Q.
\end{equation}

As in \cite{IK2}, we eliminate $v$ from the system \eqref{rhoeta} and we 
obtain the second order equation
\[
(u-\rho)u_{\rho \rho} + u_{\eta \eta} - \frac{u_{\rho}}{2} + u_{\rho}^2 =0
\]
and the system
\begin{equation} \label{system}
\begin{bmatrix}
\partial_+ u \\
\partial_- u \\
u
\end{bmatrix}_{\eta}
+
\begin{bmatrix}
-\sqrt{\rho-u} & 0 & 0 \\
\quad  0 & \sqrt{\rho-u} & 0 \\
\quad  0 & 0 & 0
\end{bmatrix}
\begin{bmatrix}
\partial_+ u \\
\partial_- u \\
u
\end{bmatrix}_{\rho}
=\begin{bmatrix}
Q(\partial_-u - \partial_+ u)\partial_+ u \\
Q(\partial_+u - \partial_- u)\partial_- u \\
\frac{1}{2}(\partial_+ u + \partial_- u)
\end{bmatrix}.
\end{equation}
We use the notation
\[
W=(\partial_+ u, \partial_-u, u)^T
\]
and in sections \S 3.2 - \S 3.4, we consider system \eqref{system} in 
subregions of the hyperbolic
interaction domain $ABCFEE_1F_1C_1B_1A$ with $u$ given on characteristic 
boundaries. We find $\partial_{\pm} u$ along those boundaries by using the 
fact that they are characteristics and by using
the characteristic decomposition equations written in the form \eqref{dec_new}.
We obtain local existence of solution in these subregions and we obtain 
certain properties of the solution and of the characteristic
curves. Our ideas are similar to those in \cite{IK2} on the study of a 
semi-hyperbolic patch.

\subsection{Interactions in the region $ABMB_1A$}

We consider system \eqref{system} in the region $ABMB_1A$. 
The boundaries $\widehat{AB}$ and
$\widehat{B_1A}$ are characteristics determined in \S 2.3 with the condition 
for $u$ found in \S 2.2.
More precisely, we have that $\widehat{AB}$ is given by
\[
\frac{(a + \eta)^2}{2} = -\xi + \frac{3a^2}{2},
\]
or, in parabolic coordinates,
\[
\rho = a^2 - a \eta - \frac{\eta^2}{4},
\]
and along $\widehat{AB}$ we have
\[
u=\xi-a\eta - a^2 = \rho - \left(\frac{\eta}{2} + a \right)^2.
\]
Also, $\widehat{B_1A}$ is given by
\[
\frac{(a-\eta)^2}{2} = -\xi + \frac{3a^2}{2} \quad \text{or } \quad 
\rho = a^2 + a \eta - \frac{\eta^2}{4},
\]
and along $\widehat{B_1A}$ we have
\[
u = \xi + a \eta - a^2 = \rho - \left(\frac{\eta}{2}-a \right)^2.
\]

We find $\partial_- u$ along $\widehat{AB}$ and $\partial_+u$ along $\widehat{B_1A}$
by using the fact that these boundaries are characteristics and we find 
$\partial_+ u$ along $\widehat{AB}$ and
$\partial_-u$ along $\widehat{B_1A}$ by using equations
\eqref{dec_new}.

Since $\widehat{AB}$ is a $\lambda_-$ characteristics, we have
\begin{equation} \label{AB-}
\partial_-u = \partial_{\eta} - \sqrt{\rho-u} \partial_{\rho} u = -\eta - 2a <0
\end{equation}
along $\widehat{AB}$. Similarly, since $\widehat{B_1A}$ is a $\lambda_+$ 
characteristics, we have
\begin{equation} \label{B1A+}
\partial_+u = u_{\eta} + \sqrt{\rho-u}\, \partial_{\rho} u = -\eta+2a >0
\end{equation}
along $\widehat{B_1A}$.

Next, since $\widehat{AB}$ is $\lambda_-$ characteristic curve, \eqref{dec_new} 
imply that along $\widehat{AB}$ we have
\[
\partial_-\big( \frac{1}{\partial_+u}\big) + \partial_-u \frac{1}{\partial_+u} =Q,
\]
i.e., by using \eqref{AB-}, we have
\[
\frac{d}{d \eta} \frac{1}{\partial_+u} - \frac{1}{\eta+2a} 
\frac{1}{\partial_+u} = \frac{1}{(\eta+2a)^2}.
\]
Notice that this is a linear ordinary differential equation in $1/\partial_+u$.
 We use \eqref{B1A+} to impose a condition at $A$,
\[
\frac{1}{\partial_+u} (A) = \frac{1}{2a},
\]
and as a solution of this initial value problem we obtain $\partial_+u$ 
along $\widehat{AB}$. Similarly,
\eqref{dec_new} imply that along $\widehat{B_1A}$ we have
\[
\frac{d}{d\eta} \frac{1}{\partial_-u} - \frac{1}{\eta - 2a} \frac{1}{\partial_-u} = \frac{1}{(\eta-2a)^2},
\]
which with the condition $\frac{1}{\partial_-u}(A)=-\frac{1}{2a}$
gives $\partial_-u$ along $\widehat{B_1A}$. Therefore, we have 
$W=(\partial_+u, \partial_-u, u)^T$
along the boundaries $\widehat{AB}$ and $\widehat{B_1A}$.

\begin{theorem}  \label{thm3.1}
The Goursat problem \eqref{system} with $W$ given on the characteristic boundaries
$\widehat{AB}$ and $\widehat{B_1A}$ as above has a smooth local solution 
at the point $A$.
\end{theorem}

\begin{proof} 
As in \cite[Theorem 4.1]{IK2}, it suffices to check the following
compatibility conditions at the point $A$:
\[
W|_{\widehat{AB}}(A)=W|_{\widehat{B_1A}}(A)
\]
and
\[
\frac{\vec{l}_3 W'|_{\widehat{AB}}(A) - \frac{\partial_+u(A)+\partial_-u(A)}{2}}
{\lambda_1(A)-\lambda_3(A)}=
\frac{\vec{l}_3 W'|_{\widehat{B_1A}}(A) - \frac{\partial_+u(A)+\partial_-u(A)}{2}}
{\lambda_2(A)-\lambda_3(A)},
\]
where $\lambda_1 = -\sqrt{\rho-u}$, $\lambda_2 = \sqrt{\rho-u}$ and $\lambda_3=0$ 
are the eigenvalues of the system \eqref{system} and $\vec{l}_1=(1,0,0)$, 
$\vec{l}_2=(0,1,0)$ and $\vec{l}_3=(0,0,1)$ are the corresponding eigenvectors. 
It is easy to check that both conditions are satisfied.
\end{proof}

\begin{lemma} \label{lem3.2}
Let $u$ be a smooth solution of the Goursat problem \eqref{system} in the 
domain $ABMB_1A$ with $W$ given on characteristic
boundaries $\widehat{AB}$ and $\widehat{B_1A}$ as above. Then
\begin{itemize}
\item[(a)] $\partial_-u < 0$ and $\partial_+ u >0$ in the interior of the 
 domain $ABMB_1A$,

\item[(b)] $\partial_-u$ is increasing along $\lambda_+$ characteristics, and

\item[(c)] $\partial_+u$ is decreasing along $\lambda_-$ characteristics.
\end{itemize}
\end{lemma}

\begin{proof} 
The proof is similar to the proof of \cite[Lemma 4.2]{IK2} and for completeness, we
outline it here.
Let $P$ be an interior point of the domain $ABMB_1A$ and let $P_1$ be the 
intersection of the $\lambda_+$
characteristics with $\widehat{AB}$. Then integrating 
\eqref{decomposition} along $\widehat{PP_1}$
we have
\[
\partial_-u(P)=\partial_-u(P_1) e ^{\int_P^{P_{1}} Q(\partial_+u - \partial_-u)ds}.
\]
Since $\partial_-u(P_1)<0$ according to \eqref{AB-}, we have $\partial_-u(P)<0$. 
Part (b) follows by noticing that $Q>0$. Similarly, we prove the statement
 (c) for $\partial_+u$.
\end{proof}

\begin{lemma} \label{lem3.3}
Let $u$ be a smooth solution of the Goursat problem \eqref{system} 
in the domain $ABMB_1A$ with $W$ given on characteristic
boundaries $\widehat{AB}$ and $\widehat{B_1A}$ as above. Then
$\Lambda_+$ characteristics are convex and $\Lambda_-$ characteristics are concave.
\end{lemma}

\begin{proof}
From \eqref{pmL} we have
\[
\partial^{\pm} \Lambda_{\pm} 
= \frac{\Lambda_{\pm}^4}{\eta \Lambda_{\pm}+2}\partial_{\pm} u.
\]
Using the result (a) of the previous Lemma, it suffices to prove
\begin{equation}\label{claim}
\eta \Lambda_{\pm}+2>0.
\end{equation}

First, we note that $\xi-u>0$ in the region $ABMB_1A$.
Clearly, at the point $A$, we have $\xi-u=a^2>0$. Assume that there is a point $P$
in the interior of $ABMB_1A$ such that $\xi-u=0$ at $P$. By the definition of 
$\Lambda_+$ in \eqref{ev_xieta}, if $\eta \neq 0$ at $P$, then $\Lambda_+$ 
characteristics would be vertical at $P$,
which is false. Hence, $\eta=0$ at $P$, and therefore $\eta^2/4=u-\xi$ at $P$, 
implying that $P$ is sonic. This is also false since the region $ABMB_1A$ 
is hyperbolic.

Next, from \eqref{ev_xieta} we have
\[
\eta \Lambda_+ + 2 = \frac{\eta^2 + \eta \sqrt{\eta^2 + 4(\xi-u)} 
+ 4(\xi-u)}{2(\xi-u)}.
\]
To prove \eqref{claim}, we need to prove that the numerator in the above 
expression is positive, i.e.,
that $\eta^2 + 4(\xi-u) > -\eta \sqrt{\eta^2 + 4(\xi-u)}$. 
If $\eta>0$, this is clearly satisfied, and if
$\eta<0$, by squaring both sides of the inequality, we obtain that the 
inequality is also satisfied.
Similarly, we prove the claim \eqref{claim} for $\Lambda_-$ characteristics.
\end{proof}

\subsection{Interactions in the simple wave region $BCDMB$}
We express the left and the right border of the rarefaction wave 
$R_{12}$ in $(\rho, \eta)$-coordinates and we obtain
\[
\rho = \left(\frac{\eta}{2}+a\right)^2 \quad \text{and} \quad
\rho = \left(\frac{\eta}{2}+a\right)^2-1.
\]
From Lemma \ref{lem3.3} we have that the $\Lambda_+$ characteristics at the point
 $B$ curves up after passing through the point $B$. Therefore,
if $\widehat{MB}$ is given by $\rho=f(\eta)$ in the $(\rho, \eta)$-coordinates,
we have $f'>0$ and $f''<\frac{1}{2}$.  Then
\[
f'=\frac{d \rho}{d \eta} = \sqrt{\rho -u} = \sqrt{f-u}
\]
implies $u=f-(f')^2$, and along $\widehat{MB}$ we have
\begin{equation} \label{sign}
\partial_+u = \partial_{\eta}u + \sqrt{\rho-u}  \partial_{\rho} u 
= f'(1-2f'')>0.
\end{equation}

Next, we prove that $u$ is continuous in the simple wave region $BCDMB$.

\begin{theorem} \label{thm3.4}
Let $u$ be a solution of the Goursat problem \eqref{system} with $W$ given 
along the characteristic boundaries $\widehat{BC}$ and $\widehat{MB}$. 
Then $\lambda_-$ characteristics do not intersect each other
in the domain $BCDMB$.
\end{theorem}

\begin{proof}
We prove the statement by contradiction. Assume that $Z \in BCDMB$ is the 
first intersection point of two negative characteristics $\overline{XZ}$ 
and $\overline{YZ}$, where $X,Y \in \widehat{MB}$. Assume
that $\partial_+u(Z)=\infty$.

Since $u$ is constant along $\lambda_-$ characteristics, we have $\partial_-u=0$ 
along $\overline{XZ}$.
From the characteristic decomposition equation \eqref{decomposition}, we have
\[
\partial_-\partial_+u = - Q (\partial_+u)^2
\]
i.e.,
\[
-\frac{\partial_-\partial_+u}{(\partial_+u)^2}=Q
\]
on $\overline{XZ}$. Integrating the last equation along the characteristic line 
$\overline{XZ}$, we obtain
\[
\frac{1}{\partial_+u(Z)}-\frac{1}{\partial_+u(X)}=\int_X^Z Q ds,
\]
implying
\[
\partial_+u(X) = \Big(-\int_X^Z Q ds\Big)^{-1} <0,
\]
since $Q>0$. This is a contradiction to \eqref{sign}.
\end{proof}

In the next lemma we prove that $u$ is increasing along $\widehat{DC}$ 
in the parabolic coordinates and decreasing in self-similar coordinates.

\begin{lemma} \label{lem3.5}
Let $u$ be a solution of the Goursat problem \eqref{system} in the domain $BCDMB$
with $W$ given along the characteristic
boundaries $\widehat{BC}$ and $\widehat{MB}$. Then
\[
\partial_+ u > 0  \text{ on }   \widehat{DC} \quad \text{and} \quad
\partial^+ u < 0  \text{ on }   \widehat{DC}.
\]
\end{lemma}

\begin{proof}
Let $Y \in \widehat{DC}$ and let $\overline{XY}$ be a negative characteristics 
through $Y$, where $X \in \widehat{MB}$. We integrate the characteristic 
decomposition equation \eqref{decomposition}
along $\overline{XY}$ and obtain
\[
\partial_+u(X) = \partial_+u(Y) e^{\int_X^Y Q(\partial_-u - \partial_+u)ds}.
\]
From \eqref{sign} we have $\partial_+u(X)>0$, implying $\partial_+u(Y)>0$.

To prove the second part of the lemma, we consider the change of coordinates
$(\xi, \eta) \mapsto (\rho, \eta)$ and we compute
\begin{align*}
\partial_+u 
& =  \partial_{\eta} u + \sqrt{\rho-u}\, \partial_{\rho}u \\
& =  -\frac{\eta}{2}\partial_{\xi} u + \partial_{\eta} u 
 + \sqrt{\rho-u}\, \partial_{\xi}u \\
& = \Big( \partial^+ u - \Lambda_+ \partial_{\eta}u\Big)
\Big( \sqrt{\rho-u} -\frac{\eta}{2}\Big) +\partial_{\eta} u,
\end{align*}
using that $\partial_{\xi} u = \partial^+u - \Lambda_+ \partial_{\eta} u$ 
from \eqref{dir_der}. By rewriting the last expression, we obtain
\[
\partial_+u = \partial^+ u \Big( \sqrt{\rho-u} - \frac{\eta}{2}\Big) 
+ \partial_{\eta} u
\Big( 1 - \Lambda_+ \Big(\sqrt{\rho-u} - \frac{\eta}{2} \Big) \Big).
\]
Using the definition of $\Lambda_+$ in \eqref{ev_xieta}, it is easy to show
\[
1 - \Lambda_+ \Big(\sqrt{\rho-u} - \frac{\eta}{2} \Big) =0.
\]
Therefore,
\[
\partial_+u = \partial^+u \Big(\sqrt{\rho-u} - \frac{\eta}{2} \Big),
\]
and to finish the proof, we need to prove that
\begin{equation} \label{claim_1}
\sqrt{\rho-u} - \frac{\eta}{2} <0 \quad \text{on }  \widehat{DC}.
\end{equation}
Clearly, since $C$ is a sonic point, we have
\[
\sqrt{\rho-u} - \frac{\eta}{2} = -\frac{\eta_C}{2} < 0,
\]
since $C$ is in the upper half-plane. If there exists a point $X \in \widehat{DC}$
 such that
\[
\sqrt{\rho-u}=\frac{\eta}{2} \quad \text{at } X,
\]
then we have $\eta = \sqrt{\eta^2 + 4(\xi-u)}$ at $X$, implying by \eqref{ev_xieta}, 
that $\Lambda_-$characteristics is horizontal at $X$. From the geometry of the 
problem, as described in \S 2.3, this
is impossible since every $\Lambda_-$ characteristics in the upper half-plane 
in the domain $BCDMB$ is decreasing. Hence, \eqref{claim_1} is proved, 
implying $\partial^+u<0$ along $\widehat{DC}$.
\end{proof}

\begin{lemma} \label{lem3.6}
Let $u$ be a solution of the Goursat problem \eqref{system} in the domain $BCDMB$
with $W$ given along the characteristic
boundaries $\widehat{BC}$ and $\widehat{MB}$. Then $\Lambda_+$ 
characteristics $\widehat{DC}$ is concave in $(\xi, \eta)$-plane.
\end{lemma}

\begin{proof}
Along $\widehat{DC}$, we have
\[
\frac{d^2 \eta}{d \xi^2} = \partial^+\Lambda_+ 
= \frac{\Lambda^3_+}{\eta \Lambda_++2} \partial^+u
\]
by \eqref{pmL}. Recall that $\Lambda_+<0$ and that $\partial^+u<0$, 
by the previous Lemma.

Next, we determine the sign of
\begin{equation}
\label{sign+}
\eta\Lambda_+ + 2 = \frac{\eta^2 + 4(\xi-u) 
+ \eta \sqrt{\eta^2 + 4(\xi-u)}}{2(\xi-u)}
\end{equation}
along $\widehat{DC}$.
We prove that
\begin{equation}
\label{sign+DC}
\eta \Lambda_+ + 2 < 0 \quad \text{on } \widehat{DC} \setminus \{C\}.
\end{equation}
First, note that at the point $C$ we have
\[
\xi-u = -5a^2 + 2a \sqrt{4a^2+2}-4<0.
\]
If there is a point $P \in \widehat{DC}$ such that $\xi-u=0$ at $P$, 
then by the definition \eqref{ev_xieta}, we would have that $\Lambda_+$ 
characteristics at $P$ is vertical, which is false.
Hence
\begin{equation}
\label{xi-u}
\xi -u < 0 \quad \text{on }  \widehat{DC}.
\end{equation}
To prove \eqref{sign+DC}, we need to show that the numerator in \eqref{sign+} 
is positive on $\widehat{DC} \setminus \{C\}$. This is clearly true as the 
sum of the first two terms is positive since $\widehat{DC} \setminus \{C\}$ 
is in the hyperbolic region, and the third term is positive since
$\eta>0$ on $\widehat{DC}$.
\end{proof}

\subsection{Interactions in the semi-hyperbolic patch $CDEFC$}

We follow the same idea as in \cite{IK2} where the semi-hyperbolic patch 
resulted from the interaction of the rarefaction wave and the sonic parabola, 
while the semi-hyperbolic patch $CDEFC$ results from the interaction of the 
simple wave and the sonic parabola. We remark again
that the solution in the semi-hyperbolic patch is coupled with the solution 
in the elliptic region through the free sonic boundary, however, this dependence 
is still an open problem. In this section, as in \cite{HW, IK2, LZ, SZ}, 
to study the solution in the region $CDEFC$, we assume that the location
of the curve $\widehat{DF}$ is known.

Let us denote the $\Lambda_+$ characteristics $\widehat{DC}$ by $\xi = f(\eta)$ and
$\Lambda_-$ characteristics $\widehat{DF}$ by $\xi=g(\eta)$. We have
$f'<0$ and, by Lemma \ref{lem3.6}, we have $f''<0$. Also, $g'<0$, and since the straight 
$\Lambda_-$ characteristics of the simple wave in the region $BCDMB$ become 
curved to the right of $\widehat{DC}$,
it is reasonable to assume that $g''>0$. As in \S 3.2, we use the fact that 
$\widehat{DC}$ and $\widehat{DF}$ are characteristics to derive $u, \partial_+u$ 
and $\partial_-u$ along $\widehat{DC}$
and $\widehat{DF}$ and we show that the Goursat problem for the system 
\eqref{system} with data given
on $\widehat{DC}$ and $\widehat{DF}$ has a smooth solution in a neighborhood of $D$.

In $(\rho, \eta)$-coordinates, we have that $\widehat{DC}$ is given by 
$\rho=f(\eta) + \eta^2/4$.
Using the fact that $\widehat{DC}$ is $\lambda_+$ characteristics, we have
\[
f'+\frac{\eta}{2} = \frac{d \rho}{d \eta} = \sqrt{\rho-u}
 = \sqrt{f+\frac{\eta^2}{4}-u},
\]
implying
\begin{equation} \label{uDC}
u=f+\frac{\eta^2}{4} - \big(f'+\frac{\eta}{2}\big)^2.
\end{equation}
Hence, along $\widehat{DC}$
\begin{equation} \label{DC}
\partial_+u  = u_{\eta} + \sqrt{\rho-u} \, u_{\rho} 
= - 2 f''\big( f'+\frac{\eta}{2}\big).
\end{equation}
We claim $f'+\frac{\eta}{2}>0$ along $\widehat{DC}$. 
At the point $C$, given by \eqref{C}, we have
\[
f'+\frac{\eta}{2}
=-\frac{2}{\eta_C} + \frac{\eta_C}{2}
=\frac{1}{a-\sqrt{4a^2+2}}-(a-\sqrt{4a^2+2})
=\frac{-5a^2-1+2a\sqrt{4a^2+2}}{a-\sqrt{4a^2+2}}.
\]
It is easy to show that both the numerator and the denominator in the above 
expression are negative, implying that the claim holds at the point $C$. 
Assume that there is a point $X \in \widehat{DC}$
such that $f'+\frac{\eta}{2}=0$ at $X$. From \eqref{uDC} and definition 
of $\widehat{DC}$ in $(\rho, \eta)$-coordinates, we have that at the point $X$
\[
u=f+\frac{\eta^2}{4}=\rho,
\]
implying that $X$ is sonic, which is false. Therefore, the above claim 
is proved and
\begin{equation} \label{DC+}
\partial_+u > 0 \quad \text{on } \widehat{DC}.
\end{equation}

Similarly, $\lambda_-$ characteristics $\widehat{DF}$ is given by 
$\rho=g(\eta)+\eta^2/4$, implying
\[
g'+\frac{\eta}{2} = \frac{d \rho}{d \eta} = - \sqrt{\rho-u} 
= - \sqrt{g(\eta)+\frac{\eta^2}{4}-u}.
\]
Hence,
\[
u=g(\eta) +\frac{\eta^2}{4} - \big( g'+\frac{\eta}{2} \big)^2,
\]
and along $\widehat{DF}$ we have
\[
\partial_-u = u_{\eta} - \sqrt{\rho-u} = -2 g''\big( g'+\frac{\eta}{2}\big).
\]
We claim that $g'+\frac{\eta}{2}<0$ on $\widehat{DF}$. Using that at the point 
$D$ we have
\[
f'+\frac{\eta}{2}= \sqrt{\rho-u} = - \big(g'+\frac{\eta}{2} \big),
\]
and we proved $f'+\frac{\eta}{2}>0$ at $D$, we obtain $g'+\frac{\eta}{2}<0$ 
at $D$. Similarly, as above
if there is a point $X \in \widehat{DF}$ such that $g'+\frac{\eta}{2}=0$ at $X$, 
then $X$ would be a sonic point. Therefore, the above claim is proved and
\begin{equation} \label{DF-}
\partial_-u > 0 \quad \text{on }  \widehat{DF}.
\end{equation}

Using \eqref{dec_new}, as in \S 3.2, we have that $\partial_u$ along 
$\widehat{DC}$ is found from a solution of the linear ordinary equation
\[
\frac{d}{d\eta} \frac{1}{\partial_-u} - \frac{f''}{2f'+\eta} \frac{1}{\partial_-u} =
\frac{1}{(2f'+\eta)^2},
\]
with the initial condition
\[
\frac{1}{\partial_-u}(D) = -\frac{1}{2g''(D)(g'(D)+\eta_D/2)},
\]
and $\partial_+u$ along $\widehat{DF}$ is found from the solution of the 
linear ordinary equation
\[
\frac{d}{d\eta} \frac{1}{\partial_+u}- \frac{g''}{2g'+\eta} \frac{1}{\partial_+u} 
= \frac{1}{(2g'+\eta)^2}
\]
with the initial condition
\[
\frac{1}{\partial_+u}(D) = - \frac{1}{2f''(D)(f'(D)+\eta_D/2)}.
\]
We obtain analogous results to those in \S 3.2.

\begin{theorem} \label{thm3.7}
The Goursat problem \eqref{system} with $W$ given on the characteristic boundaries
$\widehat{DC}$ and $\widehat{DF}$ as above has a smooth local solution at 
the point $D$.
\end{theorem}

\begin{lemma} \label{lem3.8}
Let $u$ be a smooth solution of the Goursat problem \eqref{system} in the 
domain $DCFD$ with $W$ given on characteristic
boundaries $\widehat{DC}$ and $\widehat{DF}$ as above. Then
\begin{itemize}
\item[(a)] $\partial_-u > 0$ and $\partial_+ u >0$ in the interior of 
the domain $DCFD$,

\item[(b)] $\partial_-u$ is increasing (or decreasing) along $\lambda_+$ 
characteristics if
\[
\partial_-u > \partial_+u \quad \text{(or} \quad \partial_-u < \partial_+u),
\]

\item[(c)] $\partial_+u$ is increasing (or decreasing) along $\lambda_-$ 
characteristics if
\[
\partial_+u > \partial_-u \quad  \text{(or} \quad \partial_+u < \partial_-u).
\]
\end{itemize}
\end{lemma}

\begin{remark} \label{rmk3.9} \rm
We remark that from part (a) of the previous Lemma we have that the minimum of 
$u$ in the domain $DCFD$ is achieved at the point $D$.
\end{remark}

\begin{lemma} \label{lem3.10}
Let $u$ be a smooth solution of the Goursat problem \eqref{system} in the 
domain $DCFD$ with $W$ given on characteristic
boundaries $\widehat{DC}$ and $\widehat{DF}$ as above. 
Then $\Lambda_+$ characteristics are concave
and $\Lambda_-$ characteristics are convex.
\end{lemma}

\begin{proof}
The proof is similar to the proofs of Lemmas \ref{lem3.3} and \ref{lem3.6}. 
We recall from \eqref{pmL} that
\[
\partial^{\pm} \Lambda_{\pm} = \frac{\Lambda_{\pm}^4}{\eta \Lambda_{\pm} +2} 
\partial_{\pm}u.
\]
Using the previous Lemma, it suffices to show that
\begin{equation}
\label{twosigns}
\eta \Lambda_+ + 2 < 0 \quad \text{and} \quad \eta \Lambda_-+2 >0.
\end{equation}

First recall from \eqref{xi-u} that $\xi-u<0$ on $\widehat{DC}$. 
If there is a point $P$ inside the domain $DCFD$ such that $\xi-u=0$ at $P$, 
then, by \eqref{ev_xieta}, we would have that
$\Lambda_+$ characteristics at $P$ is vertical, which is false. 
Hence $\xi-u <0$ in the domain $DCFD$.

Next, we have
\[
\eta \Lambda_{\pm} + 2 = \frac{\eta^2 + 4(\xi-u) 
\pm \eta \sqrt{\eta^2+4(\xi-u)}}{2(\xi-u)}.
\]
Clearly, $\eta^2 + 4(\xi-u) + \eta \sqrt{\eta^2+4(\xi-u)}>0$ in 
$DCFD \setminus \{C\}$, since the sum of the first two terms is positive 
because the points in $DCFD \setminus \{C\}$ are hyperbolic
and the third term is positive since $\eta>0$ in $DCFD$. 
Hence, \eqref{twosigns} holds for $\Lambda_+$.

To show that $\eta^2 + 4(\xi-u) - \eta \sqrt{\eta^2+4(\xi-u)}<0$ in 
$DCFD \setminus \{C\}$, we prove
\[
\eta^2 + 4(\xi-u) < \eta \sqrt{\eta^2 + 4(\xi-u)}.
\]
We note that the left-hand side is positive, and by squaring both sides
and recalling that $\xi-u<0$, we obtain that the inequality is satisfied.
\end{proof}


The reflected $\Lambda_+$ characteristics 
pass through $\widehat{DF}$ and continue as straight characteristics
forming a simple wave in the region $DFED$. As in \cite[\S 5]{IK2},
these characteristics form an envelope before their sonic points indicating 
existence of a shock.


\subsection*{Acknowledgments}
The second author was partially supported by the University of Houston - Downtown
Organized Research and Creative Activity Grant.

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