\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 237, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/237\hfil Riemann problem]
{Riemann problem for a two-dimensional quasilinear hyperbolic system}

\author[C. Shen \hfil EJDE-2015/237\hfilneg]
{Chun Shen}

\address{Chun Shen \newline
School of Mathematics and Statistics Science, Ludong
University, Yantai, Shandong Province 264025, China.\newline
Phone +86 535 6697510. Fax +86 535 6681264}
\email{shenchun3641@sina.com}

\thanks{Submitted April 28, 2013. Published September 15, 2015.}
\subjclass[2010]{35L65, 35L67, 76N15}
\keywords{Conservation laws; delta shock wave; Riemann  problem;
\hfill\break\indent   generalized characteristic  analysis}

\begin{abstract}
 This article concerns the study of the Riemann problem for a
 two-dimensional non-strictly hyperbolic system of conservation laws.
 The initial data  are three constant states separated by three lines
 and are chosen so that one of the three interfaces of the initial
 data projects a planar delta shock wave. Based on the generalized
 characteristic analysis, the global solutions are constructed
 completely. The solutions reveal a variety of geometric structures
 for the interactions of delta shock waves with rarefaction waves,
 shock waves and contact discontinuities.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\allowdisplaybreaks

\section{Introduction}

In this article, we study the Riemann problem of the 
two-dimensional system
\begin{equation}\label{1.1}
\begin{gathered}
 u_t+( u^2)_x+(uv)_y=0, \\
 v_{t}+(uv)_x+(v^2)_y=0,
\end{gathered}
\end{equation}
with initial data
\begin{equation} \label{1.2}
(u,v)\big|_{t=0}= \begin{cases}
(u_1,v_1), & y>0,\\
(u_2,v_2), & x<0,\; y<0, \\
(u_3,v_3), & x>0, y<0,
\end{cases}
\end{equation}
where $(u_i,v_i)$, $i=1,2,3$ are constant states. It was shown in
\cite{Chen3} that it is most suitable for the choice of initial data
as constants in each of the three sectors such as in the form
\eqref{1.2}, because it keeps the essential components of the
two-dimensional Riemann problem for a system of conservation laws
and while the number of cases is less than that in other choices.
Thus, the choice of initial data in the form \eqref{1.2} is able to
reveal the formation and development of singularity of
 solution to
the system \eqref{1.1}. Furthermore, the technique developed for the
two-dimensional Riemann problem with three constant initial data as
in  the form \eqref{1.2} can be easily generalized to other choices
of initial data. In addition, with the choice of initial data in the
form \eqref{1.2}, the restriction of wave pattern is sufficient for
deriving the expression of exact numerical fluxes such as the
positive scheme \cite{G.Wang} and the Godunov scheme \cite{L.Gosse}.

 System \eqref{1.1} can be considered as a simplification of two-dimensional
 Euler equations for it can be derived directly from the
two-dimensional isentropic Euler equations by letting the pressure
and  density be constants in the last two momentum equations
\cite{D.Tan1}. In fact, the simplified system \eqref{1.1} is also
able to explain some interesting phenomena in gas dynamics such as
diffractions along wedges \cite{D.Tan1}. The system \eqref{1.1} also
belongs to the system of type
\begin{equation}\label{1.3}
\begin{gathered}
 u_t+(uf(u,v))_x+(ug(u,v))_y=0,\\
 v_{t}+(vf(u,v))_x+(vg(u,v))_y=0,
\end{gathered}
\end{equation}
where $f(u,v)=u$ and $g(u,v)=v$. Equations like \eqref{1.3} occur in
a variety of applications, including oil recovery, elastic theory
and magneto-hydrodynamics \cite{Chen1}. The Riemann problem for
\eqref{1.3} is much more complicated than the scalar case, but it is
simpler than the problem for general hyperbolic system. This is due
to the fact that the domain of mixed type does not appear in the
study of self-similar solutions for \eqref{1.3}. Thus the study of
the Riemann problem for \eqref{1.3} can be regarded as a necessary
step to more complicated and practical cases such as the conjectures
on the two-dimensional Riemann solutions for the Euler equations
\cite{ZZ}. When $f(u,v)=g(u,v)=u$ and $v=\rho$, the Riemann problem
for the system \eqref{1.3} was investigated in
\cite{ShenSunWang,Sun1}.

There have been many studies on system \eqref{1.1} from various
aspects. Tan and Zhang \cite{D.Tan1} firstly studied the four
quadrant Riemann problem  of \eqref{1.1}, namely the initial data
are four constant states in each quadrant of $(x,y)$ plane, and they
discovered that a kind of new nonlinear wave called delta shock wave
was there. About the delta shock wave solution in the
multi-dimensional hyperbolic conservation laws, we can also see
\cite{ GSZ,LSZ,J.Li1,J.Li2,Yang1,Yang2,
V.M.Shelkovich,W.Sheng,Sun,Zheng} and the reference therein. Yang
and Zhang \cite{YZ} verified the analytic solutions in \cite{D.Tan1}
numerically using the MmB preserving scheme.  Lopes-Filho and
Nussenzveig Lopes \cite{Lopes-Filho} have investigated the
singularity formation about the evolution along a characteristic of
the compression rate of nearby characteristics for \eqref{1.1}. Wang
\cite{Wang} proposed an example to show that the solution for
\eqref{1.1} is not unique. The three constant Riemann problem for
\eqref{1.1}, that is the initial data take three constant states in
three angular domains in the $(x,y)$ plane, was studied in
\cite{Chen1,Chen2,Yang1,Yang2,Yang3}. Huang and Yang \cite{F.Huang}
constructed the solutions for the two constant Riemann problem of
\eqref{1.1}.  The non-selfsimilar Riemann problem on (1.1) was also
considered by  Chen, Wang and Yang \cite{G.Q.Chen} and they
discovered the triple-shock pattern there.

Our goal in this article is to construct explicitly the global
solutions for \eqref{1.1} and \eqref{1.2}. The initial data as
\eqref{1.2} simplify the complexity of the structures of the four
quadrant Riemann solutions, while the essential ingredients of
two-dimensional Riemann problems can hold. Following \cite{D.Tan1},
we assume that the initial data \eqref{1.2} are chosen so that only
one planar elementary wave appears at each interface of the initial
data. The justification of this choice lies in that the majority of
physical observations only involve the study of a single propagation
wave type \cite{W.Hwang}. The most attractive feature of the system
\eqref{1.1} is in that the plane delta shock wave appears in the
solutions of Rieamnn problem \eqref{1.1} and \eqref{1.2} for some
certain initial data.  Thus, we draw our attention on the cases that
one of the three planar elementary waves is a planar delta shock
wave, which are different from the previous results. Using the
method of generalized characteristic analysis, we solve the Riemann
problem \eqref{1.1} and \eqref{1.2} analytically and nine exact
entropy solutions with different geometric structures are
constructed globally. The solutions reveal various interactions of
delta shock waves with the classical waves involving contact
discontinuities, shock waves and rarefaction waves. The  evolution
of the planar delta shock wave is presented in detail. The results
of the present note provide a preparation of theoretical analysis
for the numerical simulation for \eqref{1.1}.


The rest of this article  is organized as follows. In section 2, we
provide some basic properties of system \eqref{1.1}  for
completeness, including the characteristics, bounded discontinuities
and delta shock waves. In section 3, we classify the Riemann problem
according to the combinations of the exterior waves. Then the global
solutions are constructed by the method of generalized
characteristic analysis.


\section{Preliminaries}

In this section, we briefly review some basic properties of system
\eqref{1.1}  for readers' convenience, and the detailed
 study can be found in \cite{D.Tan1}.

Since both  \eqref{1.1} and \eqref{1.2} are invariant under the
self-similar transformation $(t,x,y)\to (\alpha t,\alpha x,\alpha y)$
 with $\alpha>0$, we seek the self-similar solution of
the form $ (u,v)(t,x,y)=(u,v)(\xi,\eta)$ where
$(\xi,\eta)=(x/t,y/t)$. The system for this form of solution is
\begin{equation}\label{2.1}
\begin{gathered}
 -\xi u_\xi-\eta  u_\eta+( u^2)_\xi+(uv)_\eta=0,
\\
 -\xi v_{\xi}-\eta  v_\eta+(uv)_\xi+(v^2)_\eta=0,
\end{gathered}
\end{equation}
and the initial data \eqref{1.2} become boundary values at infinity
\begin{equation} \label{2.2}
\lim_{\xi^2+\eta^2\to\infty}(u,v)= \begin{cases}
 (u_1,v_1), & \eta>0,\\
 (u_2,v_2), & \xi<0, \eta<0, \\
 (u_3,v_3), & \xi>0, \eta<0.
\end{cases}
\end{equation}
System \eqref{2.1} has two eigenvalues
\begin{equation} \label{2.3}
\lambda_1=\frac{v-\eta}{u-\xi},\quad
\lambda_2=\frac{2v-\eta}{2u-\xi},
\end{equation}
which are called pseudo-characteristics of \eqref{1.1} for a given
solution $(u,\rho)(\xi,\eta)$. The $\lambda_1$ pseudo-characteristic
field is linearly degenerate and the $\lambda_2$
pseudo-characteristic field is genuinely nonlinear if
$u\eta-v\xi\neq0$.


Define the characteristic curves $\Gamma_i $ $(i=1,2)$ in the
$(\xi,\eta)$ plane by
\begin{equation}\label{2.4}
\Gamma_i: \frac{d\eta}{d\xi}=\lambda_i.
\end{equation}
The singularity point for $\Gamma_i$, denoted by $P_i$, is
$P_i=(iu,iv)$, $i=1,2$.  We call the curve $\eta/\xi=v/u$ the base
curve denoted by $B$, which consists of singularity points for
$\Gamma_i$ and the degenerate hyperbolic points
$\lambda_1=\lambda_2$. We stipulate the direction of characteristic
curves $\Gamma_i$ from infinity to the singularity point $P_i$
$(i=1,2)$, which is motivated by virtue of the increase of time
\cite{J.Li2}.

(i) Smooth solution.
 If $v/u=\rm{constant}$ in some domain, we call it a simple wave of
 the second kind, which satisfies $u_\xi+\lambda_2u_\eta=0$. The
 simple wave is called a rarefaction wave (abbr. $R$),  if all the
 $\lambda_2-$ characteristic curves and their extensions in the
 positive directions do not intersect until they reach the
 corresponding base curve.

(ii) Bounded discontinuity solution.
Let $\eta=\eta(\xi)$ be a smooth discontinuity of a bounded
discontinuous solution in the $(\xi,\eta)$ plane. Solving the
Rankine-Hugoniot condition, we obtain the following two kinds of
discontinuities.

 A contact discontinuity (abbr. $J$) satisfies
\begin{equation}\label{2.5}
\frac{d\eta}{d\xi}=\sigma_1=\frac{\eta-v_+}{\xi-u_+}=\frac{\eta-v_-}{\xi-u_-},
\end{equation}
which is the  $\lambda_1-$ characteristic line for both sides.
Hereafter, $(u_\pm,v_\pm)$ represent the limit states on two sides
of the discontinuity $\eta=\eta(\xi)$.

 A shock wave (abbr. $S$) satisfies
\begin{equation}\label{2.6}
\frac{d\eta}{d\xi}=\sigma_2=\frac{\eta-(v_++v_-)}{\xi-(u_++u_-)},
\quad \frac{v_+}{u_+}=\frac{v_-}{u_-},
\end{equation}
and the  entropy condition which can be defined as  ``three incoming,
one outgoing'', that is, at any point of the discontinuity, three of
the characteristic lines, two $\Gamma_2$s and one $\Gamma_1$, come
into the point and the remaining one, $\Gamma_1$, goes out.
Similarly to characteristic curves, we orient the integral curve of
$d\eta/d\xi=\sigma_i$ to point towards the singularity point
$(\xi,\eta)=(u_++(i-1)u_-,v_++(i-1)v_-)$, $i=1,2$.



(iii) Delta shock wave.
A discontinuity in $(u,v)(\xi,\eta)$ at $\xi=\xi(\eta)$ is called a
delta shock wave (abbr. $\delta$) if it satisfies
\begin{equation}\label{2.7}
\frac{d\xi}{d\eta}=\frac{\xi-(u_++u_-)}{\eta-(v_++v_-)},
\end{equation}
and the entropy condition which can be defined as ``none outgoing'',
that means that all of the characteristic lines on both sides of the
discontinuity curve do not go out at every point of the
discontinuity. Similarly, the direction of a delta shock wave is
towards its singular point $(\xi,\eta)=(u_++u_-,v_++v_-)$.

\section{Construction of Solutions involving one $\delta$}


We consider now the Riemann problem \eqref{1.1} and \eqref{1.2}. It
is obvious that,  outside a sufficiently large circle in the
$(\xi,\eta)$ plane, the solution must be constant states connected
by three one-dimensional planar waves $(u,v)(\xi)$ or $(u,v)(\eta)$,
which are called planar elementary waves or exterior waves. In this
note, we deal with the cases in which exactly one of the three
one-dimensional waves from infinity is a delta shock wave. We assume
first that the exterior wave connecting states $(u_2,v_2)$ and
$(u_3,v_3)$ is a delta shock wave $\delta_{23}$, so that $u_2>0>u_3$
should be satisfied. According to the remaining two exterior waves,
we find that there exist five different combinations which lead to
topologically distinct solutions. The combinations are as follows:
 1. $R_{12}\delta_{23}R_{31}$,  \quad 2. $R_{12}\delta_{23}S_{31}$, \quad 3. 
 $R_{12}\delta_{23}J_{31}$,\quad
 4. $S_{12}\delta_{23}J_{31}$, \quad 5. $J_{12}\delta_{23}J_{31}$.

What we need to do in the following is to extend the exterior
solutions inwards to construct our global Riemann solutions. We will
deal with this problem case by case according to the above
classification. Here and below, $\delta_{ij}$ denotes the delta
shock wave with $(u_i,v_i)$ and  $(u_j,v_j)$ on its two sides, also
for $R_{ij}$, $S_{ij}$, $J_{ij}$.
\smallskip

\noindent\textbf{Case 1.}  $R_{12}\delta_{23}R_{31}$
The occurrence of this case depends on the condition: $v_2<v_1$,
$v_3<v_1$ and $u_1/v_1=u_2/v_2=u_3/v_3$, where the value of
$u_1/v_1$ has two possibilities: $u_1/v_1>0$ or $u_1/v_1<0$. We only
need to construct the solution for $u_1/v_1>0$ since the other case
can be treated in the same way.

By the theory of Cauchy problems, we know that the determination
domain of the constant state $(u_3,v_3)$ is
$\Omega_1=\{(\xi,\eta)\big|\xi>u_2+u_3, \eta<2v_3\}$, namely,
$(u,v)(\xi,\eta)=(u_3,v_3)$ when $(\xi,\eta)\in \Omega_1$. So
$\delta_{23}$ will stay straight until it meets the point
$(\xi_0,\eta_0)=(u_2+u_3,2v_3)$.

Let $R_{30}$ (resp. $R_{10}$) denote the part of $R_{31}$ where the
$v$-component of the solution satisfies $v_3\leq v<0$ (resp. $0\leq
v\leq v_1$). Then $\delta_{23}$ will penetrate $R_{30}$ to form a
new delta shock wave $\delta_{2R}:$ $\xi=\xi(\eta)$ which satisfies
\begin{equation}\label{3.1}
\begin{gathered}
 \frac{d\xi}{d\eta}=\frac{\xi-(u+u_{2})}{\eta-(v+v_{2})},\\
 \eta=2v, \\
\frac{u}{v}=\frac{u_2}{v_2},\quad  v_3\leq v< 0, \\
\xi_0=u_2+u_3, \quad  \eta_0= 2v_3.
\end{gathered}
\end{equation}
From this equation, we find that the tangent line of this
discontinuity always points to the singularity points
$(\xi,\eta)=(u+u_{2},v+v_{2})$. Therefore the integral curve of
\eqref{3.1} is convex. Substituting $v=\eta/2$, $u=u_2\eta/2v_2$
into the first equation in \eqref{3.1} yields
\begin{equation}\label{3.2}
 \frac{d\xi}{d\eta}=\frac{2\xi-2u_{2}-u_2\eta/v_2}{\eta-2v_{2}}.
\end{equation}
With the initial condition $(\xi_0,\eta_0)=(u_2+u_3, 2v_3)$ in mind,
an easy calculation leads to
\begin{equation}\label{3.3}
 \xi-2u_2=\frac{u_2}{v_2}(\eta-2v_2)-\frac{u_2}{4v_2(v_3-v_2)}(\eta-2v_2)^2.
\end{equation}

The delta shock wave $\delta_{2R}$ cannot cancel the whole
rarefaction wave $R_{31}$ and it ends at the point
$(\xi_1,\eta_1)=(u_2v_2/(v_2-v_3),0)$, where a shock wave $S_{2R}$
develops by the  ``three incoming, one outgoing'' entropy condition.
The shock wave penetrates part of the rarefaction wave $R_{10}$ and
it has the same expression as \eqref{3.3}, namely the curve of
$S_{2R}$ is the continuation of that of $\delta_{2R}$.  It can be
found from \eqref{3.3} that $d \xi/d \eta\to u_2/v_2$ as
$\eta\to 2v_2$ which means that $S_{2R}$ vanishes
tangentially to the point $(2u_2,2v_2)$.

\begin{figure}[htb]
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\put(76.63,34.88){\makebox(0,0)[cc]{\footnotesize${R_{30}}$}}
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\put(22,60.75){\makebox(0,0)[cc]{\footnotesize${R_{12}}$}}
\put(48.63,67.25){\makebox(0,0)[cc]{\footnotesize${\rlap{$\bigcirc$}\
{\mbox{\tiny $1$}}}$}}%
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\put(64.13,20.75){\makebox(0,0)[cc]{\footnotesize${\rlap{$\bigcirc$}\
{\mbox{\tiny $3$}}}$}}%
 \put(21.38,20.5){\makebox(0,0)[cc]{\footnotesize${\rlap{$\bigcirc$}\ {\mbox{\tiny $2$}}}$}}
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\caption{Solution for Case 1 when $u_1/v_1>0$.}
\label{fig1}
\end{figure}

We illustrate the global structure of the solution in Figure \ref{fig1}.  For
convenience, we use some notations in the following figures.  $(i)$,
$(\bar{i})$,  $(i+j)$, represent points $(\xi,\eta)=(u_i,v_i)$,
$(\xi,\eta)=(2u_i,2v_i)$, $(\xi,\eta)=(u_i+u_j,v_i+v_j)$,
respectively. And ${\rlap{$\bigcirc$}\ {\mbox{\tiny $i$}}}$ stands
for the state $(u_i,v_i)$.
\smallskip


\noindent\textbf{Case 2.}  $R_{12}\delta_{23}S_{31}$
 This case happens if and only if $v_2<0<v_1<v_3$,
$u_3<u_1<0<u_2$ and $u_1/v_1=u_2/v_2=u_3/v_3$.


The construction of solution for this case is analogous to that in
Case 1. The difference lies in that the  shock wave $S_{R3}$
penetrates the whole rarefaction wave $R_{10}$ and ends at the point
$(\xi_0,2v_1)$ with the slope
$$\frac{d\eta}{d\xi}=\frac{\eta-(v_1+v_3)}{\xi-(u_1+u_3)},$$
where $\xi_0$ can be obtained by substituting $\eta=2v_1$ into
$$
 \xi-2u_3=\frac{u_3}{v_3}(\eta-2v_3)-\frac{u_3}{4v_3(v_2-v_3)}(\eta-2v_3)^2.
$$
Thereafter the shock wave stays straight  with $(u_1,v_1)$ and
$(u_3,v_3)$ as the limit states on two sides until it matches with
$S_{31}$ at the singularity point $(u_1+u_3,v_1+v_3)$. See Figure \ref{fig2}.

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\caption{ Solution for Case 2.}
\label{fig2}
\end{figure}

\smallskip


\noindent\textbf{Case 3.}  $R_{12}\delta_{23}J_{31}$
The appearance of this case depends on the conditions $v_2<v_1$,
$v_1=v_3$ and $u_1/v_1=u_2/v_2$. The discussion for this case can be
further divided into three subcases according to the values of
$u_1/v_1$ and $v_1$: a. $u_1/v_1>0$; b. $u_1/v_1<0$ and $v_1<0$; c.
$u_1/v_1<0$ and $v_1>0$.
\smallskip


\noindent\textbf{Subcase 3a.}  $u_1/v_1>0$.
Without loss of generality, we assume that $v_1<2v_2$. Since the
determination domain of constant state $(u_3,v_3)$ is
$\Omega_2=\{(\xi,\eta)\big|\xi>u_2+u_3, \eta<v_3\}$, it follows that
$(u,v)(\xi,\eta)=(u_3,v_3)$ when $(\xi,\eta)\in \Omega_2$. By the
fact that $J_{31}$ intersects the base curve of constant $(u_1,v_1)$
only at the point $(\xi,\eta)=(u_1,v_1)$, $J_{31}$ will stay
straight until it meets the point $(\xi,\eta)=(u_2+u_3,v_3)$. So we
have
$$
\lim_{\eta\to
J_{31}+0}(u(\xi,\eta),v(\xi,\eta))=(u_1,v_1).$$ Solving the boundary
value problem at the point $(u_2+u_3,v_3)$ with the boundary
conditions
$$
\lim_{\xi\to
\delta_{23}-0}(u(\xi,\eta),v(\xi,\eta))=(u_2,v_2), \quad
\lim_{\eta\to J_{31}+0}(u(\xi,\eta),v(\xi,\eta))=(u_1,v_1),
$$
we find that a shock wave, denoted by $S_{21}$, is the solution.
Here and in what follows, $\xi\to \delta_{ij}-0$ (resp.
$\delta_{ij}+0$) means that for any point $(\xi_0,\eta_0)\in
\delta_{ij}$, $(\xi,\eta)\to(\xi_0,\eta_0)$ with $\xi<\xi_0$
(resp. $\xi>\xi_0$). The similar notation is $\eta\to
\delta_{ij}\pm0$.


The shock wave $S_{21}: \eta-v_3=v_2(\xi-u_2-u_3)/(u_1-u_3)$ cannot
keep straight after it meets the rarefaction wave $R_{12}$. Then the
shock wave $S_{R1}$ begins to cancel $R_{12}$ and stops tangentially
at the point $(2u_1,2v_1)$. See Figure \ref{fig3}.

\begin{figure}[htb] 
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\caption{Solution for Subcase 3a.}
\label{fig3}
\end{figure}

\begin{figure}[htb]
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\caption{Solution for Subcase 3b.}
\label{fig4}
\end{figure}

\smallskip

\noindent\textbf{Subcase 3b.}  $u_1/v_1<0$ and $v_1<0$.
It is obvious that $(u,v)(\xi,\eta)=(u_2,v_2)$ when  $\xi<u_2+u_3$
and  $\eta<2v_2$. The delta shock wave $\delta_{23}$ cannot arrive
at its singularity point $(u_2+u_3,v_2+v_3)$ for the reason that it
will interact with  $R_{12}$. The
 interaction gives rise to a new delta shock wave $\delta_{R3}:\xi=\xi(\eta)$ 
 which will
 penetrate $R_{12}$ and has a varying speed expressed as
\begin{equation}\label{3.4}
\begin{gathered}
 \frac{d\xi}{d\eta}=\frac{\xi-(u+u_{3})}{\eta-(v+v_{3})},\\
 \eta=2v, \\
\frac{u}{v}=\frac{u_2}{v_2},\quad  \ v_2\leq v\leq v_1 \\
\xi_0=u_2+u_3, \quad \eta_0= 2v_2.
\end{gathered}
\end{equation}
A similar calculation as in Case 1 leads to
\begin{equation}\label{3.5}
 \xi-v_3(\frac{u_3}{v_3}+\frac{u_2}{v_2})
 =\frac{u_2}{v_2}(\eta-2v_3)-\frac{u_2}{4v_2(v_2-v_3)}(\eta-2v_3)^2.
\end{equation}
It can be found that the curve $\xi=\xi(\eta)$ lies below the line
$\xi=u_3+u_2(\eta-v_3)/v_2$ which consists of singularity points for
the curve. In fact, in view of $u_2/v_2<0$ and $v_2<v_3$, it can be
derived from \eqref{3.5} that
$$
\xi-u_3(\frac{u_3}{v_3}+\frac{u_2}{v_2})<\frac{u_2}{v_2}(\eta-2v_3),
$$
which gives $\xi<u_3+u_2(\eta-v_3)/v_2$. Therefore $\delta_{R3}$ is
able to cancel the whole $R_{12}$ completely and disappears
tangentially at the point $(u_1+u_3,v_1+v_3)$.

The contact discontinuity $J_{31}$ can go straight until it reaches
its singularity point $(u_3,v_3)$ where a delta shock wave
$\delta_{13}$ should be constructed to separate the states
$(u_3,v_3)$ and $(u_1,v_1)$. Finally $\delta_{13}$ matches with
$\delta_{R3}$ at its singularity point $(u_1+u_3,v_1+v_3)$. See 
Figure \ref{fig4}.
\smallskip



\noindent\textbf{Subcase 3c.}  $u_1/v_1<0$ and $v_1>0$.
The discussion for the interaction of $\delta_{23}$ and  $R_{20}$ is
the same as that in Case 3b. At the point
$(u_3-v_3u_2/(v_2-v_3),0)$, the delta shock wave is decomposed into
a contact discontinuity  $J_{43}$ and a shock wave $S_{4R}$ with the
intermediate state $(u_4,v_4)$ between them. Here $(u_4,v_4)$
denotes the crossing point of the base curve of constant $(u_1,v_1)$
and $\delta_{R3}$'s tangent line at the point
$(u_3-v_3u_2/(v_2-v_3),0)$ which passes through the  point
$(u_3,v_3)$.

The contact discontinuity $J_{43}$ connecting states $(u_4,v_4)$ and
$(u_3,v_3)$ matches with $J_{31}$ at their common singularity point
$(u_3, v_3)$ where we find a centered rarefaction wave $R_{41}$ is
the solution by solving the boundary value problem with the boundary
conditions
$$
\lim_{\xi\to J_{43}-0}(u(\xi,\eta),v(\xi,\eta))=(u_4,v_4),
\quad \lim_{\eta\to
J_{31}+0}(u(\xi,\eta),v(\xi,\eta))=(u_1,v_1).
$$
 Then the shock wave $S_{4R}$ which connects the states on $R_{10}$ and
 $(u_4,v_4)$ must interact with $R_{41}$,  penetrate it and finally terminate at the point
 $(2u_1,2v_1)$. The shock wave and rarefaction wave are the second
 kind of waves, so their interaction can be obtained similarly to
 the situation of the scalar conservation law. See Figure \ref{fig5}.


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%\put(49,7.75){\makebox(0,0)[cc]{Figure \ref{fig5}. Solution for Subcase 3c.}}
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\caption{Solution for Subcase 3c.}
\label{fig5}
\end{figure}

\smallskip



\noindent\text{Case 4}  $S_{12}\delta_{23}J_{31}$.
This case occurs when $v_2>v_1$, $v_2.v_1>0$, $v_1=v_3$ and
$u_1/v_1=u_2/v_2$ are satisfied. We also proceed our discussion
through two subcases according to the value of $u_1/v_1$: a.
$u_1/v_1>0$; b. $u_1/v_1<0$.
\smallskip

\noindent\textbf{Subcase 4a.} $u_1/v_1>0$.
Similarly to Case 3a, the collision of  $\delta_{23}$
 and $J_{31}$ happens at the point $(u_2+u_3,v_3)$ where a
 rarefaction wave $R_{21}$ if the point $(u_2+u_3,v_3)$ lies above the line
 $\eta=v_1/u_1\xi$ or a shock wave $S_{21}$ otherwise should be constructed.
When $R_{21}$ appears, the exterior wave $S_{12}$ will penetrate it
and finally end at the point $(2u_1,2v_1)$. See Figure \ref{fig6}. If $S_{21}$
forms, the exterior wave $S_{12}$ can go straight until it arrives
at its singularity point $(u_1+u_2,v_1+v_2)$ which is  also the
ending point of $S_{21}$.

\begin{figure}[htb]
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\caption{Solution involving $R_{21}$ for  Subcase 4a.}
\label{fig6}
\end{figure}

\smallskip

\noindent\textbf{Subcase 4b.} $u_1/v_1<0$.
It is clear to see that  $(u,v)(\xi,\eta)=(u_2,v_2)$ when
$\xi<u_2+u_3$ and  $\eta<v_2+v_3$. The exterior waves $\delta_{23}$
and $J_{31}$ can arrive at  their singularity points respectively,
while the shock wave $S_{12}$ cannot and it stops at the point
$(u_2+u_3,v_2+v_3)$. Then a delta shock wave $\delta_{31}:
\eta=v_3+u_1/v_1(\xi-u_3)$ should be constructed to separate the
states $(u_1,v_1)$ and $(u_3,v_3)$ lying between $J_{31}$ and
$S_{12}$, the ending point of which is also  $(u_2+u_3,v_2+v_3)$.
 See Figure \ref{fig7}.

\begin{figure}[htb]
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\caption{Solution for Subcase 4b.}
\label{fig7}
\end{figure}

\smallskip

\noindent\textbf{Case 5.}  $J_{12}\delta_{23}J_{31}$.
In this case, the initial data satisfy $v_1=v_2=v_3$. There are two
subcases corresponding to topologically distinct solutions in view
of the sign of $u_1/v_1$.

\smallskip

\noindent\textbf{Subcase 5a.} $u_1/v_1>0$.
 The three exterior waves collide at the point $(u_2+u_3,v_3)$,
 where the solution in the region $\{(\xi,\eta)\big| \eta>v_3\}$ is a
 shock wave $S_{\infty1}$ penetrates a centered rarefaction wave
 $R_{1\infty}$. Here the rarefaction wave $R_{1\infty}$ connects the
 states $(u_1,v_1)$ and infinity, also for $S_{\infty1}$. See Figure
 \ref{fig8}

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\caption{Solution for Subcase 5a.}
\label{fig8}
\end{figure}

\smallskip


\noindent\textbf{Subcase 5b.} $u_1/v_1<0$.
Different from the above subcase, only two exterior waves $J_{12}$
and $J_{31}$ interact for this subcase, while the exterior wave
$\delta_{23}$ can keep straight and arrive at its singularity point
$(u_2+u_3,v_2+v_3)$. The interaction of the two contact
discontinuities results in a delta shock wave $\delta_{32}$
connecting the states $(u_2,v_2)$ and $(u_3,v_3)$. Such a delta
shock wave $\delta_{32}$ is not unique, the expression of which may
be any line starting from any point $(\xi,v_3)$ where $u_3<\xi<u_2$
and ending at the point $(u_2+u_3,v_2+v_3)$. So the solution for
this subcase is not unique. See Figure \ref{fig9}.

\begin{figure}[htb]
\begin{center}
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\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi % GNUPLOT compatibility
\begin{picture}(112.98,67)(-9,12) %(112.98,70.08)(-25,5)
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\put(30.05,49.85){\circle*{.8}}%
\put(28.15,51.94){\makebox(0,0)[cc]{$\scriptstyle (3)$}}
\put(61.82,49.85){\circle*{.8}}%
\put(63.82,51.94){\makebox(0,0)[cc]{$\scriptstyle (2)$}}
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\put(42.06,27.28){\makebox(0,0)[cc]{$\scriptstyle (3+2)$}}
\put(46.84,19.71){\vector(0,1){.07}}
\put(43.0,39.07){\vector(1,-3){.07}}
\put(23.08,49.85){\vector(1,0){.07}}
\put(69.93,49.88){\vector(-1,0){.07}}
\put(35.46,49.88){\vector(1,0){.07}}
\put(45.18,49.85){\vector(-1,0){.07}}
\textcolor[rgb]{0.00,1.00,0.00}{\put(18.31,52.71){\makebox(0,0)[cc]{\footnotesize${J_{12}}$}}}
\put(52.51,49.85){\circle*{.8}}%
\put(49.51,51.84){\makebox(0,0)[cc]{$\scriptstyle (1)$}}
\textcolor[rgb]{0.00,1.00,0.00}{\put(74.86,52.77){\makebox(0,0)[cc]{\footnotesize${J_{31}}$}}}
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\textcolor[rgb]{1.00,0.00,0.00}{\put(50.31,18.61){\makebox(0,0)[cc]{\footnotesize${\delta_{23}}$}}}
\put(68.97,30.69){\makebox(0,0)[cc]{\footnotesize${\rlap{$\bigcirc$}\
{\mbox{\tiny $3$}}}$}}%
\textcolor[rgb]{1.00,0.00,0.00}{\put(45.92,41.31){\makebox(0,0)[cc]{\footnotesize${\delta_{32}}$}}}
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\put(9.79,66.22){\line(1,0){4.415}}
\put(9.79,70.32){\vector(0,1){.07}}
\put(9.79,66.12){\line(0,1){4.205}}
\put(5.95,69.59){\makebox(0,0)[cc]{\footnotesize${\eta,v}$}}
\put(17.59,66.22){\makebox(0,0)[cc]{\footnotesize${\xi,u}$}}
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{\mbox{\tiny $1$}}}$}}%
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\put(25.87,60.55){\makebox(0,0)[cc]{\footnotesize${\rlap{$\bigcirc$}\
{\mbox{\tiny $1$}}}$}}%
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%\put(49,7.75){\makebox(0,0)[cc]{Fig. 9. Solution for Subcase 5b. }}
\end{picture}
\end{center}
\caption{Solution for Subcase 5b.}
\label{fig9}
\end{figure}

So far, we have finished the construction of solutions to the
Riemann problem \eqref{1.1} and \eqref{1.2} when the exterior wave
connecting states $(u_2,v_2)$ and $(u_3,v_3)$ is a delta shock wave
$\delta_{23}$, and the other two exterior waves are classical waves
as shock waves, rarefaction waves and contact discontinuities.  The
formation and evolution of singularities in the solutions of Riemann
problem \eqref{1.1} and \eqref{1.2} are analyzed in details, which
is a major difficulty in solving hyperbolic systems of conservation
laws. For the other cases when the exterior waves involves only one
delta shock wave propagating along in the $x-$ direction, two delta
shock waves, or three delta shock waves, the discussion is
complicated and we will study them in the future.

\subsection*{Acknowledgments}
 This work is partially supported by National
Natural Science Foundation of China (11441002,11271176) and Shandong
Provincial Natural Science Foundation (ZR2014AM024).

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\end{document}