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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 320, pp. 1--25.\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-2016/320\hfil Structural stability of Riemann solutions]
{Structural stability of Riemann solutions for strictly
hyperbolic systems with three piecewise constant states}

\author[X. Wei, C. Shen \hfil EJDE-2016/320\hfilneg]
{Xuefeng Wei, Chun Shen}

\address{Xuefeng Wei \newline
School of Mathematics and Statistics Science,
Ludong University,
Yantai, Shandong Province 264025, China}
\email{50816179@qq.com}

\address{Chun Shen (corresponding author)\newline
School of Mathematics and Statistics Science,
Ludong University,
Yantai, Shandong Province 264025, China}
\email{shenchun3641@sina.com}

\thanks{Submitted September 22, 2016. Published December 14, 2016.}
\subjclass[2010]{35L65, 35L67, 76N15}
\keywords{Delta shock wave; wave interaction; Riemann problem;
\hfill\break\indent strict hyperbolicity}

\begin{abstract}
 This article concerns the wave interaction problem for a
 strictly hyperbolic system of conservation laws whose Riemann
 solutions involve delta shock waves. To cover all situations,
 the global solutions are constructed when the initial data are taken
 as three piecewise constant states. It is shown that the Riemann
 solutions are stable with respect to a specific small perturbation of
 the Riemann initial data.
 In addition, some interesting nonlinear phenomena are captured during
 the process of constructing the solutions, such as the generation and
 decomposition of delta shock waves.
\end{abstract}

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

\section{Introduction}

In this article, we are concerned with the hyperbolic system of
conservation laws
\begin{equation}\label{e1.1}
\begin{gathered}
u_t+(u^2)_x=0, \\
v_t+\Big((2u+1)v\Big)_x=0,
\end{gathered}
\end{equation}
which was used to study the behavior of a
magnetohydrodynamics model (MHD) \cite{D.Tan1,D.Tan2}. System \eqref{e1.1}
is strictly hyperbolic whose eigenvalues are $\lambda_1=2u$ and
$\lambda_2=2u+1$. Furthermore, the first characteristic field for
$\lambda_1$ is genuinely nonlinear and the second one for
$\lambda_2$ is linearly degenerate.


The Riemann problem is the particular Cauchy problem with the two
piecewise initial data
\begin{equation} \label{e1.2}
(u,v)(x,0)= \begin{cases}
(u_{-},v_{-}),& x<0,\\
(u_{+},v_{+}),& x>0,
\end{cases}
\end{equation}
where all the $u_\pm$ and $v_\pm$ are given constants. The Riemann
problem \eqref{e1.1} and \eqref{e1.2} was studied by Tan \cite{D.Tan1}
through the self-similar vanishing viscosity method. It was
discovered in \cite{D.Tan1} that if $u_{+}<u_{-}-1$, then the
 solution of the Riemann problem \eqref{e1.1} and \eqref{e1.2} cannot
be constructed by a combination of shock waves, rarefaction waves and contact
discontinuities. In this situation, the delta shock wave should be
introduced into the soluton, which is the form of the standard
Dirac delta function supported on a shock wave
\cite{G.Q.Chen,M.Colombeau,B.L.Keyfitz,B.L.Keyfitz2,J.Li,W.Sheng,H.Yang1}.

With the Riemann solutions of \eqref{e1.1} and \eqref{e1.2} in hand, it is natural
to expect the study of the so-called double Riemann problem because
the Riemann problem \eqref{e1.1} and \eqref{e1.2} cannot describe
the dynamic pictures in all the situations for \eqref{e1.1}. In
this article, we study the Cauchy problem for \eqref{e1.1} with three
piecewise initial data:
\begin{equation}\label{e1.3}
(u,v)(x,0)= \begin{cases}
 (u_{-}, v_{-}), & -\infty<x< -\varepsilon, \\
 (u_{m},v_{m}), & -\varepsilon<x<\varepsilon, \\
 (u_{+}, v_{+}), & \varepsilon< x<+\infty,
 \end{cases}
\end{equation}
where $\varepsilon>0$ is arbitrarily small. This is the so-called
perturbed Riemann problem or the double Riemann problem. For the
reason that the initial data \eqref{e1.3} may be regarded as a small
perturbation of the corresponding Riemann initial data \eqref{e1.2} with
the perturbed parameter $\varepsilon$. In fact, we will encounter an
interesting problem that if the limits $\varepsilon\to 0$ of
 solutions $(u_{\varepsilon},v_{\varepsilon})(x,t)$ are identical
with the ones of the Riemann problem \eqref{e1.1} and \eqref{e1.2} or not,
in which $(u_{\varepsilon},v_{\varepsilon})(x,t)$ refer to the
 solutions of the particular Cauchy problem \eqref{e1.1} and \eqref{e1.3}
associated with $\varepsilon$ accordingly.

In fact, the three piecewise initial data \eqref{e1.3} have been widely
used to study the wave interaction problem for some hyperbolic systems, such as
the the pressureless Euler system \cite{C.Shen2,H.Yang2}, the Euler
system for Chaplygin gas \cite{L.Guo2,A.Qu}, a non-strictly
hyperbolic system \cite{C.Shen4,A.Yao} and various types of
chromatography systems \cite{L.Guo1,C.Shen1,M.Sun}. It is noticed
that all the systems studied above belong to the so-called Temple
class \cite{B.Temple}, namely the shock curves coincide with the
rarefaction curves in the phase plane, such that wave interactions
have relatively more simplified structures and then the global
solutions may be constructed completely for these systems with the
initial data \eqref{e1.3}. However, it is remarkable that the system \eqref{e1.1}
does not belong to the Temple class, such that the solutions of the
perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3} have more complicated and
interesting structures. Fortunately, we discover that the
propagation speeds of elementary waves for the Riemann problem \eqref{e1.1}
and \eqref{e1.2} can be expressed concisely by the state variable $u$,
including shock wave, rarefaction wave, contact discontinuity and
delta shock wave. Thus, the global solutions of the perturbed
Riemann problem \eqref{e1.1} and \eqref{e1.3} can be constructed in explicit
forms.

The main purpose of this paper is to investigate various possible
 wave interactions including delta shock waves for system
\eqref{e1.1}. Thus, we take the three piecewise initial data \eqref{e1.3} instead
of the Riemann initial data \eqref{e1.2} such that the solutions beyond the
interactions are constructed. Furthermore, it is shown that the
solutions of the perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3} converge
to the corresponding ones of the Riemann problem \eqref{e1.1} and \eqref{e1.2} as
$\varepsilon\to0$ by dealing with this problem case by
case, which shows the stability of Riemann solutions with respect to
the small perturbation \eqref{e1.3} of the Riemann initial data \eqref{e1.2}. In
addition, some interesting nonlinear phenomena can be captured
during the process of constructing the solutions to the perturbed
Riemann problem \eqref{e1.1} and \eqref{e1.3}. At first, we discover that a delta
shock wave may be generated by the interaction between two shock
waves. Secondly, it can be observed that a delta shock wave may be
decomposed into a shock wave and a delta contact discontinuity
during the process when it penetrates a rarefaction wave. Finally,
it can be shown that infinitely many contact discontinuities may be
continuously produced which have the same propagation speed during
the process when a shock wave penetrates a rarefaction wave.

It should be pointed out that the following strictly hyperbolic
system of conservation laws
\begin{equation}\label{e1.4}
\begin{gathered}
u_t+\Big(\frac{u^2}{2}\Big)_x=0, \\
v_t+\Big((u-1)v\Big)_x=0,
\end{gathered}
\end{equation}
was introduced by Hayes and LeFloch \cite{B.T.Hayes}. This system has the
similar property with system \eqref{e1.1}. It should be stressed that
the interactions between the delta shock wave with the other
elementary waves have been well investigated for \eqref{e1.4} by
Nedeljkov and Oberguggenberger \cite{M.Nedeljkov4}. 
The method of split delta function \cite{M.Nedeljkov1,M.Nedeljkov2,M.Nedeljkov3}
was in \cite{M.Nedeljkov4} to study the strength of delta shock wave precisely.

In this article, the wave interaction problem
is also considered when the delta shock wave does not appear at the initial
moment for the perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3},
 which was not addressed in \cite{M.Nedeljkov4}.
In fact, the interactions between the delta shock wave with the other elementary
waves for \eqref{e1.1} have similar structures with those
for \eqref{e1.4}. In this paper, we only use the generalized
Rankine-Hugoniot conditions to calculate the strength of delta shock
wave for simplicity. In addition, the stability of solutions to the
Riemann problem \eqref{e1.1} and \eqref{e1.2} can also be analyzed when the delta
shock waves are involved in the solutions to the perturbed Riemann
problem \eqref{e1.1} and \eqref{e1.3}.


This article is organized in the following way. In Section 2, some
preliminaries are given, which include the Riemann solutions of
\eqref{e1.1} and \eqref{e1.2} and the generalized Rankine-Hugoniot relations of
 delta shock wave. Furthermore, it is proven rigorously that the
delta shock wave solution indeed satisfies the system \eqref{e1.1} in the
sense of distributions.
In Section 3, we consider the perturbed
Riemann problem \eqref{e1.1} and \eqref{e1.3} when the delta shock wave does not
appear at the initial time. The wave interaction problems are
studied in detail and then the global solutions are constructed
completely.
In Section 4, we consider the perturbed Riemann problem
\eqref{e1.1} and \eqref{e1.3} when the delta shock wave is involved at the initial
time. The interactions between the delta shock wave with the other
elementary waves are investigated carefully, including shock wave,
rarefaction wave and contact discontinuity. At the end,
 discussions are carried out and the
conclusions are drawn in Section 5.

\section{The Riemann problem}

In this section, we are devoted to the Riemann problem \eqref{e1.1} and
\eqref{e1.2}, which was investigated in \cite{D.Tan1} by using the
self-similar vanishing viscosity method. The eigenvalues of
system \eqref{e1.1} are $\lambda_1=2u$ and $\lambda_2=2u+1$, thus
 \eqref{e1.1} is strictly hyperbolic for the reason that
$\lambda_1<\lambda_2$ holds for any $u$. Furthermore, the
corresponding right eigenvectors are $r_1=(-1,2v)^T$ and
$r_2=(0,1)^T$, respectively. Thus, we have
$\nabla\lambda_1\cdot r_1=-2$ and $\nabla\lambda_2\cdot r_2=0$,
in which the symbol $\nabla$ expresses the gradient with respect to $(u,v)$.
We know that the first characteristic field for $\lambda_1$ is genuinely
nonlinear and the second one for $\lambda_2$ is always linearly
degenerate. Therefore, the waves of the first family are either
rarefaction waves (denoted by $R$) or shock waves (denoted by $S$)
which are decided by the initial data and while the waves of the
second family are always contact discontinuities (denoted by $J$).

 We first consider the elementary wave for \eqref{e1.1}.
For a given left state $(u_{-},v_{-})$, the 1-rarefaction wave curve
in the $(u,v)$ phase plane can be expressed as $R(u_{-},v_{-})$:
\begin{equation}\label{e2.1}
\begin{gathered}
\xi=\lambda_1=2u, \\
v\cdot e^{2u}=v_{-}\cdot e^{2u_{-}}, \\
u>u_{-}, \quad 0<v<v_{-}.
\end{gathered}
\end{equation}
On the other hand, the 1-shock wave curve in the $(u,v)$ phase plane
can also be expressed as $S(u_{-},v_{-})$:
\begin{equation}\label{e2.2}
\begin{gathered}
\sigma=u_{-}+u, \\
\frac{v}{v_{-}}=\frac{u_{-}-u+1}{u-u_{-}+1}, \\
u_{-}-1<u<u_{-}, \quad v>v_{-}.
\end{gathered}
\end{equation}
In addition, the 2-contact discontinuity curve in the $(u,v)$ phase
plane should satisfy $u=u_{-}$ and the corresponding propagation
speed is $\tau=2u_{-}+1=2u+1$.

Then, we construct the Riemann solutions of \eqref{e1.1} and \eqref{e1.2} for
different cases. For the case $u_{-}<u_{+}$, the Riemann solution of
\eqref{e1.1} and \eqref{e1.2} is a rarefaction wave followed by a contact
discontinuity, which can be expressed as
\begin{equation}\label{e2.3}
(u, v)(x,t)= \begin{cases}
(u_{-},v_{-}), & x<2u_{-}t, \\
\Big(\frac{x}{2t},v_{-}\exp(2u_{-}-\frac{x}{t})\Big), & 2u_{-}t<x<2u_{+}t,\\
(u_{+},v_{-}\exp(2u_{-}-2u_{+})), & 2u_{+}t<x<(2u_{+}+1)t,\\
(u_{+},v_{+}), & x>(2u_{+}+1)t.
 \end{cases}
\end{equation}
For the case $u_{-}-1<u_{+}<u_{-}$, the Riemann solution of \eqref{e1.1}
and \eqref{e1.2} contains a shock wave plus a contact discontinuity, which
is given by
\begin{equation}\label{e2.4}
(u, v)(x,t)= \begin{cases}
(u_{-},v_{-}), & x<(u_{-}+u_{+})t, \\
(u_{+},v_{*}), & (u_{-}+u_{+})t<x<(2u_{+}+1)t,\\
(u_{+},v_{+}), & x>(2u_{+}+1)t,
 \end{cases}
\end{equation}
where
\begin{equation}\label{e2.5}
v_{*}=v_{-}\cdot\frac{u_{-}-u_{+}+1}{u_{+}-u_{-}+1}.
\end{equation}

\begin{figure}[ht]
\begin{center}
\unitlength 0.9mm
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi % GNUPLOT compatibility
\begin{picture}(77,45)(47,0)
\put(47.09,39.88){\vector(0,1){.07}}
\put(47.06,33.12){\line(0,1){6.76}}
\put(47.47,41.52){\makebox(0,0)[cc]{$\scriptstyle v$}}
\put(123,6){\vector(1,0){.07}} \put(46.5,6){\line(1,0){76.5}}
\put(124.25,7.38){\makebox(0,0)[cc]{$\scriptstyle u$}}
\textcolor[rgb]{0.00,0.00,1.00}{\bezier{500}(87.5,6)(87.5,24.315)(87.5,42.63)}
\bezier{60}(69.88,6)(69.88,24.315)(69.88,42.63)
\textcolor[rgb]{0.00,0.00,1.00}{\put(89.75,40.88){\makebox(0,0)[cc]{\footnotesize$J$}}}
\put(69.88,4){\makebox(0,0)[cc]{\footnotesize$u_--1$}}
\put(87.55,4){\makebox(0,0)[cc]{\footnotesize$u_-$}}
\put(99.25,25.13){\makebox(0,0)[cc]{\footnotesize I\!I}}
\put(54,23.88){\makebox(0,0)[cc]{\footnotesize I\!I\!I}}
\textcolor[rgb]{1.00,0.00,0.00}{\put(72.75,40.88){{\makebox(0,0)[cc]{\footnotesize$S$}}}}
\put(115,8.88){\makebox(0,0)[cc]{\footnotesize$R$}}
\put(76.38,12.5){\makebox(0,0)[cc]{\footnotesize I}}
\qbezier(87.5,12.63)(97.5,7.25)(116.5,6.88)
\textcolor[rgb]{1.00,0.00,0.00}{\qbezier(70.5,42.5)(70.94,20.19)(87.5,12.63)}
\put(87.5,12.63){\circle*{.6}}
\put(94,13){\makebox(0,0)[cc]{\footnotesize$(u_-,v_-)$}}
\end{picture}
\end{center}
\caption{The $(u,v)$ phase plane for system \eqref{e1.1}
for a given left state $(u_-,v_-)$.}
\label{fig1}
\end{figure}


Let us turn our attention on the case $u_{+}\leq u_{-}-1$.
Then the nonclassical situation appears where the Riemann
problem \eqref{e1.1} and \eqref{e1.2} cannot be solved by a combination of shock
waves, rarefaction waves and contact discontinuities. To
solve the Riemann problem \eqref{e1.1} and \eqref{e1.2} when $u_{+}\leq u_{-}-1$, it
was shown in \cite{D.Tan1} that a solution containing a weighted
$\delta$-measure supported on a curve should be adopted. In this
paper, let us use the exact definition of
 delta shock wave solution which was introduced by Danilov
and Shelkovich \cite{V.G.Danilov1,V.G.Danilov2,V.G.Danilov3} and
improved by Kalisch and Mitrovic \cite{H.Kalisch1,H.Kalisch2}.

\begin{definition} \label{def2.1} \rm
Using the the two-dimensional weighted $\delta$-measure $\beta(s)\delta_{\Gamma}$,
which is supported on a smooth curve $\Gamma=\{(x(s),t(s)):a<s<b\}$,
we define the measure-valued solutions by
\begin{equation}\label{e2.6} \rm
\langle\beta(s)\delta_\Gamma,\psi(x,t)\rangle=\int_{a}^{b}\beta(s)\psi(x(s),t(s))ds,
\end{equation}
for all $\psi(x,t)\in C^{\infty}_0(R\times R_+)$.
\end{definition}

 Let $\Gamma=\{\gamma_{i}|i\in I\}$ be a set of curves in the closed
 upper half-plane $\{(x,t)|(x,t)\in (-\infty,\infty)\times
 [0,\infty)\}$, in which $\gamma_{i}$ expresses a Lipschitz continuous curve
and $I$ is a finite index set. Let  $I_0$ be the subset of $I$ involving
all the indices of curves starting from the $x-$axis and then let
$\Gamma_0=\{x_{j}^{0}|j\in I_0\}$
 be the set of initial points of all the curves $\gamma_{j}$ when $j\in
 I_0$. In what follows, one may define the
 solutions in the distributional sense to the Cauchy problem
 for the system \eqref{e1.1} with the delta measure
initial data.


\begin{definition} \label{def2.2}\rm
Let $(u,v)$ be a pair of distributions, in which $v$
 has the form
\begin{equation} \label{e2.7}
v(x,t)=\hat{v}(x,t)+\beta(x,t)\delta(\Gamma)=\hat{v}(x,t)+\sum_{i\in
I}\alpha_{i}(x,t)\delta(\gamma_{i}),
\end{equation}
and $u,\hat{v} \in L^{\infty}(R\times R_{+})$. Let us consider the
 initial data of type
\begin{equation} \label{e2.8}
(u,v)(x,0)=\Big(u_0(x),\hat{v}_0(x)+\sum_{j\in I_0}
\alpha_{j}(x^{0}_{j},0)\delta(x-x^{0}_{j})\Big),
\end{equation}
in which $u_0,\hat{v}_0\in L^{\infty}(R)$, then the above pair
of distributions $(u,v)$ are called as a generalized delta shock
wave solution of the Cauchy problem \eqref{e1.1} and \eqref{e2.8} if the following
integral equalities hold for any $\psi\in C_{c}^{\infty}(R\times R_{+})$:
\begin{equation} \label{e2.9}
  \int_{R_{+}} \int_{R}\left( u \psi _t+u^{2}%
\psi_x\right) \,dx\,dt+ \int_{R} u_0(x)\psi(x,0) dx=0,
\end{equation}
and
\begin{equation} \label{e2.10}
\begin{aligned}
&\int_{R_{+}} \int_{R}\left( \hat{v}\psi _t+(2u+1)\hat{v}%
\psi _x\right) \,dx\,dt
+\sum_{i\in I} \int_{\gamma_{i}}\alpha_{i}(x,t)\frac{\partial
\psi(x,t) }{\partial l} \\
&+ \int_{R} \hat{v}_0(x)\psi(x,0) dx+\sum_{k\in I_0}
\alpha_{k}(x^{0}_{k},0)\psi(x^{0}_{k},0)=0,
\end{aligned}
\end{equation}
where  $\int_{\gamma_{i}}$ is the line integral along
 $\gamma_{i}$ and $\frac{\partial \psi }{\partial l}$ is the
tangential derivative of $\psi$.
\end{definition}

In view of the above definitions, we use the following theorem to
describe the delta shock wave solution to the Riemann problem \eqref{e1.1}
and \eqref{e1.2} when $u_{+}\leq u_{-}-1$.

\begin{theorem} \label{thm2.3}
 For the case $u_{+}\leq u_{-}-1$, the Riemann solution of \eqref{e1.1}
and \eqref{e1.2} is piecewise smooth in the form
\begin{equation}\label{e2.11}
(u, v)(x,t)= \begin{cases}
(u_{-},v_{-}), & x<\sigma_{\delta} t, \\
(u_{\delta},\beta(t)\delta(x-\sigma_{\delta} t)), &x=\sigma_{\delta} t,\\
(u_{+},v_{+}), & x>\sigma_{\delta} t,
 \end{cases}
\end{equation}
where
\begin{equation}\label{e2.12}
\begin{gathered}
 \sigma_{\delta}=u_{-}+u_{+},\quad u_{\delta}=\frac{1}{2}(u_{-}+u_{+}-1), \\
\beta(t)=\Big((u_{-}-u_{+})(v_{-}+v_{+})-(v_{+}-v_{-})\Big)t.
\end{gathered}
\end{equation}
The measure-valued solution \eqref{e2.11} should satisfy
the generalized Rankine-Hugoniot conditions
\begin{equation}\label{e2.13}
\begin{gathered}
\frac{dx}{dt}=\sigma_{\delta},\\
\frac{d\beta(t)}{dt}=\sigma_{\delta}[v]-[(2u+1)v],\\
\sigma_{\delta}[u]=[u^2],
 \end{gathered}
\end{equation}
and the over-compressive entropy condition
\begin{equation}\label{e2.14}
2u_{+}+1<\sigma_{\delta}<2u_{-},
\end{equation}
where $[u]=u(x(t)+0,t)-u(x(t)-0,t)$ denotes the jump of $u$ across
the discontinuity $x=x(t)$, etc.
\end{theorem}

\begin{proof}
Let us check that the measure-valued solution \eqref{e2.11} with \eqref{e2.12}
should satisfy the system \eqref{e1.1} in the sense of distributions.
In other words, we need to check that \eqref{e2.11} and \eqref{e2.12} should satisfy
\begin{equation} \label{e2.15}
\begin{gathered}
 \int_0^{\infty }  \int_{-\infty }^{\infty }\left( u\psi _t+u^2
\psi _x\right) \,dx\,dt=0,  \\
 \int_0^{\infty }  \int_{-\infty }^{\infty}
\left( v\psi _t+(2u+1)v \psi_x\right) \,dx\,dt=0,
\end{gathered}
\end{equation}
for any $\psi\in C_{c}^{\infty}(R\times R_{+})$. Without loss of
generality, let us assume that $\sigma_{\delta}>0$ for the reason
that $\sigma_{\delta}\leq0$ can be dealt with similarly and the difference
only lies in that the different integral regions
are decomposed in the upper-half physical plane $(x,t) \in (R\times R_{+})$.

Let us check the first equation in \eqref{e2.15}. By  the third
equation in \eqref{e2.13}, we have
\begin{align*}
&\int_0^{\infty }\int_{-\infty }^{\infty }\left( u\psi _t+u^2
\psi _x\right) \,dx\,dt \\
&=\int_0^{\infty }\int_{-\infty }^{\sigma_{\delta} t}\left(
u_{-}\psi _t+ u_{-}^2\psi _x\right) \,dx\,dt+\int_0^{\infty
}\int_{\sigma_{\delta} t}^{\infty }\left( u_{+}\psi _t+u_{+}^2\psi
_x\right) \,dx\,dt
\\
&=\int_0^{\infty }\int_{-\infty }^{0}u_{-}\psi
_t\,dx\,dt+\int_0^{\infty }\int_0^{\sigma_{\delta} t}u_{-}\psi
_t\,dx\,dt+\int_0^{\infty}\int_{-\infty}^{\sigma_{\delta}
t}u_{-}^2\psi_x \,dx\,dt \\
&\quad +\int_0^{\infty }\int_{\sigma_{\delta}
t}^{\infty }\left( u_{+}\psi _t+u_{+}^2\psi_x\right) \,dx\,dt \\
&=\int_0^{\infty }\int_{\frac{x}{\sigma_{\delta}}}^{\infty
}u_{-}\psi _t\,dt\,dx+\int_0^{\infty }\int_{-\infty
}^{\sigma_{\delta} t}u_{-}^2\psi_x
\,dx\,dt+\int_0^{\infty}\int_0^{\frac{x}{\sigma_{\delta}}}u_{+}\psi
_t\,dt\,dx \\
&\quad +\int_0^{\infty }\int_{\sigma_{\delta} t}^{\infty
}u_{+}^2\psi_x\,dx\,dt \\
&=-\int_0^{\infty }u_{-}\psi
(x,\frac{x}{\sigma_{\delta}})dx+\int_0^{\infty
}u_{-}^2\psi(\sigma_{\delta} t,t) dt+\int_0^{\infty }u_{+}\psi
(x,\frac{x}{\sigma_{\delta}})dx \\
&\quad -\int_0^{\infty}u_{+}^2\psi(\sigma_{\delta} t,t) dt \\
&= \int_0^{\infty
}\Big(\sigma_{\delta}(u_{+}-u_{-})-(u_{+}^2-u_{-}^2)\Big)\psi(\sigma_{\delta}
t,t)dt
=0,
\end{align*}
in which we have used the fact that $\psi(x,t)$ is compactly support
in the region $R\times R_{+}$.

On the other hand, taking into account the second equation in
\eqref{e2.13} and the relation formula $\sigma_{\delta}=2 u_{\delta}+1$
from \eqref{e2.12}, we also have
\begin{align*}
&\int_0^{\infty }\int_{-\infty }^{\infty }\left( v\psi _t+(2u+1)v%
\psi _x\right) \,dx\,dt \\
&=\int_0^{\infty }\int_{-\infty }^{\sigma_{\delta} t}\left(
v_{-}\psi _t+(2u_{-}+1)v_{-}\psi _x\right) \,dx\,dt \\
&\quad +\int_0^{\infty}\int_{\sigma_{\delta} t}^{\infty }\left( v_{+}\psi
_t+(2u_{+}+1)v_{+}\psi _x\right) \,dx\,dt \\
&\quad+\int_0^{\infty}\beta(t)(\psi_t(\sigma_{\delta} t,t)+(2u_{\delta}+1) \psi_x(\sigma_{\delta} t,t))dt\\
&=\int_0^{\infty }\int_{0 }^{\sigma_{\delta} t}v_{-}\psi_t\,dx\,dt
+\int_0^{\infty}\int_{-\infty}^{\sigma_{\delta}
t}(2u_{-}+1)v_{-}\psi_x \,dx\,dt \\
&\quad +\int_0^{\infty }\int_{\sigma_{\delta}
t}^{\infty } v_{+}\psi _t\,dx\,dt+\int_0^{\infty
}\int_{\sigma_{\delta}
t}^{\infty }(2u_{+}+1)v_{+}\psi_x \,dx\,dt \\
&\quad +\int_0^{\infty}\beta(t)(\psi_t(\sigma_{\delta} t,t)+\sigma_{\delta} \psi_x(\sigma_{\delta} t,t))dt\\
&=\int_0^{\infty }\int_{\frac{x}{\sigma_{\delta}}}^{\infty
}v_{-}\psi _t\,dt\,dx
+\int_0^{\infty }\int_{-\infty }^{\sigma_{\delta} t}(2u_{-}+1)v_{-}\psi_x
\,dx\,dt \\
&\quad +\int_0^{\infty}\int_0^{\frac{x}{\sigma_{\delta}}}v_{+}\psi_t\,dt\,dx
 +\int_0^{\infty }\int_{\sigma_{\delta} t}^{\infty}(2u_{+}+1)v_{+}
 \psi_x\,dx\,dt \\
&\quad +\int_0^{\infty}\beta(t)d\psi(\sigma_{\delta} t,t)\\
&= \int_0^{\infty }(v_{+}-v_{-})\psi(x,\frac{x}{\sigma_{\delta}})dx
 +\int_0^{\infty}((2u_{-}+1)v_{-}-(2u_{+}+1)v_{+})\psi(\sigma_{\delta} t,t) dt \\
&\quad +\int_0^{\infty}\beta(t)d\psi(\sigma_{\delta} t,t) \\
&=\int_0^{\infty}\Big(\sigma_{\delta}(v_{+}-v_{-})
 +(2u_{-}+1)v_{-}-(2u_{+}+1)v_{+}-\beta'(t)\Big)\psi(\sigma_{\delta} t,t)dt
=0.
\end{align*}

It is easy to check that the measure-valued solution \eqref{e2.11} with \eqref{e2.12}
can be derived from the generalized Rankine-Hugoniot
 conditions \eqref{e2.13} by a simple calculation.
In order to ensure uniqueness, the $\delta-$entropy condition
$\lambda_1(u_{r})<\lambda_2(u_{r})<\sigma<\lambda_1(u_{l})<\lambda_2(u_{l})$
should be satisfied, which leads to the over-compressive entropy
condition \eqref{e2.14}. In other words, all the characteristics on both
sides of the $\delta-$shock wave curve are incoming. Thus, it can be
concluded from the above calculations that \eqref{e2.11} with \eqref{e2.12} is
indeed the piecewise smooth solution of the Riemann problem \eqref{e1.1}
and \eqref{e1.2} in the sense of distributions when $u_{+}\leq u_{-}-1$.
\end{proof}

\begin{remark} \label{rmk2.4} \rm
One can see that the Riemann solutions of \eqref{e1.1} and \eqref{e1.2}
can be constructed by a combination of shock waves, rarefaction waves,
contact discontinuities and delta shock waves. More precisely, there
are exactly three configurations of the Riemann solutions of \eqref{e1.1}
and \eqref{e1.2} according to the relation between $u_{-}$ and $u_{+}$ as
follows: $R+J$ when $u_{+}>u_{-}$, $S+J$ when $u_{-}-1<u_{+}<u_{-}$
and $\delta S$ when $u_{+}\leq u_{-}-1$.
\end{remark}

\section{Interactions of classical waves}

To study the perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3} is in essence
to study the wave interaction problem for the system \eqref{e1.1}. It is
remarkable that the Riemann solution of \eqref{e1.1} and \eqref{e1.2} may contain
 the delta shock wave or not. In order to cover all the cases completely,
our discussion should be divided into two parts according to the appearance of
delta shock wave or not at the initial time. In this section, we are
mainly concerned with the wave interaction problem which does not
involve the delta shock wave at the initial time. Then, we have four
possibilities according to the different combinations of the
classical waves from $(-\varepsilon,0)$ and $(\varepsilon,0)$ as
follows:
(1) $S+J$ and $S+J$; \quad (2) $R+J$ and $S+J$; \quad(3) $S+J$ and
$R+J$;\quad(4) $R+J$ and $R+J$.
\smallskip


\noindent\textbf{Case 3.1: $S+J$ and $S+J$.}
First of all, let us consider the situation that both a shock wave followed
by a contact discontinuity emit from the initial points
$(-\varepsilon,0)$ and $(\varepsilon,0)$ respectively (see Figure \ref{fig2}).
The occurrence of this case depends on the conditions
$u_{-}-1<u_{m}<u_{-}$ and $u_{+}<u_{m}<u_{+}+1$, from which we can
easily get $u_{+}<u_{-}$. The propagation speed of the first contact
discontinuity $J_1$ is $\tau_1=2u_{m}+1$ and that of the second
shock wave $S_2$ is $\sigma_2=u_{m}+u_{+}$. Thus, it is easy to
see that $J_1$ overtakes $S_2$ in finite time. The intersection
$(x_1,t_1)$ is determined by
\begin{equation}\label{e3.1}
\begin{gathered}
 x_1+\varepsilon=(2u_{m}+1)t_1, \\
 x_1-\varepsilon=(u_{m}+u_{+})t_1,
 \end{gathered}
\end{equation}
which implies
\begin{equation}\label{e3.2}
 (x_1,t_1)=\Big(\frac{\varepsilon(3u_{m}+u_{+}+1)}{u_{m}-u_{+}+1},
\frac{2\varepsilon}{u_{m}-u_{+}+1}\Big).
\end{equation}

It can be shown that a new local Riemann problem for the system
\eqref{e1.1} will be formulated at the intersection $(x_1,t_1)$ with
the initial data
\begin{equation}\label{e3.3}
(u,v)(x,t_1)=\begin{cases}
 (u_1,v_1),& x< x_1, \\
 (u_2,v_2),& x>x_1,
 \end{cases}
\end{equation}
where
\begin{gather}\label{e3.4}
(u_1,v_1)=\Big(u_{m},-v_{-}\cdot\frac{u_{m}-u_{-}-1}{u_{m}-u_{-}+1}\Big), \\
\label{e3.5}
(u_2,v_2)=\Big(u_{+},-v_{m}\cdot\frac{u_{+}-u_{m}-1}{u_{+}-u_{m}+1}\Big).
\end{gather}
To solve the new Riemann problem \eqref{e1.1} and \eqref{e3.3}, one can see that a
new shock wave followed by a new contact discontinuity will be
generated after the interaction between $J_1$ and $S_2$. Let us
denote them by $S_3$ and $J_3$ respectively. One can see that
the propagation speeds of $S_3$ and $J_3$ are the same as those
of $S_2$ and $J_2$ respectively for the reason that
$u_1=u_{m}$ and $u_2=u_{+}$.

Then, $S_3$ and $S_1$ intersect at the point $(x_2,t_2)$,
which can be calculated by
\begin{equation}\label{e3.6}
\begin{gathered}
 x_2+\varepsilon=(u_{-}+u_{m})t_2, \\
 x_2-\varepsilon=(u_{m}+u_{+})t_2.
 \end{gathered}
\end{equation}
An easy calculation leads to
\begin{equation}\label{e3.7}
(x_2,t_2)=\Big(\frac{\varepsilon(2u_{m}+u_{+}+u_{-})}{u_{-}-u_{+}},
\frac{2\varepsilon}{u_{-}-u_{+}}\Big).
\end{equation}

As before, a new local Riemann problem for  system \eqref{e1.1}
will also be formulated at the intersection $(x_2,t_2)$ with the
initial data
\begin{equation}\label{e3.8}
(u,v)(x,t_2)= \begin{cases}
 (u_{-},v_{-}),& x<x_2, \\
 (u_3,v_3),& x>x_2,
 \end{cases}
\end{equation}
in which
\begin{equation}\label{e3.9}
(u_3,v_3)=\Big(u_{+},-v_1\cdot\frac{u_{+}-u_{m}-1}{u_{+}-u_{m}+1}\Big).
\end{equation}

\begin{figure}[ht]
\begin{center}
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\end{center}
\caption{Interactions between $S+J$ and $S+J$
for two different situations where both
$u_{-}-1<u_{m}<u_{-}$ and $u_{+}<u_{m}<u_{+}+1$ should be satisfied.}
\label{fig2}
\end{figure}

It is easy to get $u_--2<u_+<u_-$ from $u_{-}-1<u_{m}<u_{-}$ and
$u_{+}<u_{m}<u_{+}+1$. In what follows, we can obtain two different
situations according to the values $u_+$ and $u_{-}-1$, which may be
described by using the lemma below.

\begin{lemma} \label{lem3.1}
For the local Riemann problem \eqref{e1.1} and \eqref{e3.8}, if
$u_--2<u_{+}<u_{-}-1$,  then a new delta shock wave will be generated; 
otherwise, if $u_{-}-1<u_{+}<u_{-}$, then there is also a new shock wave followed
by a new contact discontinuity. Finally, the global solutions of the
perturbed Riemann problem \eqref{e1.1} and \eqref{e3.1} can be illustrated in
Figure 2, in which both $u_{-}-1<u_{m}<u_{-}$ and
$u_{+}<u_{m}<u_{+}+1$ should be satisfied.
\end{lemma}

\begin{proof}
If $u_--2<u_{+}<u_{-}-1$, then a delta shock wave $\delta S$ is
generated after the coalescence of $S_1$ and $S_3$ at the point
$(x_2,t_2)$ (see Figure \ref{fig2}(a)), whose propagation speed and strength
can be calculated by the generalized Rankine-Hugoniot
 conditions \eqref{e2.13}. Thus, one can easily get that the propagation
 speed of $\delta S$ is $\sigma_{\delta}=u_{-}+u_{+}$.
Consequently, the delta shock wave $\delta S$ propagates with an
invariant speed $u_{-}+u_{+}$ for the reason that
$u_3=u_2=u_{+}$. On the other hand, the strength changes at the
different growth rates due to the differences among $v_3$, $v_2$
and $v_{+}$.

In what follows, the delta shock wave $\delta S$ intersects $J_3$
and consequently $J_2$ in finite time. The intersection
$(x_3,t_3)$ of $\delta S$ and $J_3$ can be calculated by
\begin{equation}\label{e3.10}
\begin{gathered}
 x_3-x_2=(u_{-}+u_{+})(t_3-t_2), \\
 x_3-x_1=(2u_{+}+1)(t_3-t_1),
 \end{gathered}
\end{equation}
which implies 
\begin{equation}\label{e3.11}
\begin{gathered}
\begin{aligned}
x_3&=\frac{\varepsilon(u_++u_--2u_m)(u_{+}+u_{-})}{(u_{-}-u_{+})(u_{-}-u_{+}-1)}+\frac{\varepsilon(3u_{m}-3u_{+}-1)(u_{+}+u_{-})}
{(u_{m}-u_{+}+1)(u_{-}-u_{+}-1)} \\
&\quad+\frac{\varepsilon(2u_{m}-u_{+}-u_{-})}{u_{-}-u_{+}},
\end{aligned}\\
t_3=\frac{\varepsilon(u_{-}+u_{+}-2u_{m})}{(u_{-}-u_{+})(u_{-}-u_{+}-1)}+\frac{\varepsilon
(3u_{m}-3u_{+}-1)}{(u_{m}-u_{+}+1)(u_{-}-u_{+}-1)}.
 \end{gathered}
\end{equation}
After $\delta S$ passes through $(x_3,t_3)$, it still propagates
with the invariant speed $u_{-}+u_{+}$ and overtakes $J_2$ at the
intersection $(x_4,t_4)$, which is determined by
\begin{equation}\label{e3.12}
 \begin{gathered}
 x_4-x_2=(u_{-}+u_{+})(t_4-t_2), \\
 x_4-\varepsilon=(2u_{+}+1)t_4,
 \end{gathered}
\end{equation}
such that we have
\begin{equation}\label{e3.13}
(x_4,t_4)=\Big(\frac{2\varepsilon(2u_{+}+1)(u_{-}
-u_{m})}{(u_{-}-u_{+})(u_{-}-u_{+}-1)}+\varepsilon,
\frac{2\varepsilon(u_{-}-u_{m})}{(u_{-}-u_{+})(u_{-}-u_{+}-1)}\Big).
\end{equation}
After the time $t_4$, the delta shock wave passes through $J_2$
and moves forwards with the same speed as before.

Thus, the strength of delta shock wave $\delta S$ can be calculated
respectively by
\begin{gather}\label{e3.14}
\beta(t)=\Big((u_{-}-u_{+})(v_{-}+v_3)-(v_3-v_{-})\Big)(t-t_2),
\quad \text{for } t_2\leq t \leq t_3, \\
\label{e3.15}
\beta(t)=\beta(t_3)+\Big((u_{-}-u_{+})(v_{-}+v_2)-(v_2-v_{-})\Big)(t-t_3),
\quad \text{for } t_3<t \leq t_4, \\
\label{e3.16}
\beta(t)=\beta(t_4)+\Big((u_{-}-u_{+})(v_{-}+v_{+})-(v_{+}-v_{-})\Big)(t-t_4),
\quad \text{for } t>t_4.
\end{gather}

Otherwise, if $u_{-}-1<u_{+}<u_{-}$, then a new shock wave (denotes
by $S_4$) followed by a new contact discontinuity (denotes by
$J_4$) appears after the time $t_2$ (see Figure \ref{fig2}(b)). 
The propagation speed of $S_4$ is $\sigma_4=u_{-}+u_{+}$, which
leads to $\sigma_3<\sigma_4<\sigma_1$ for $u_{+}<u_{m}<u_{-}$.
The propagation speed of $J_4$ is still $2u_{+}+1$, which is equal
to those of $J_3$ and $J_2$. It is clear that the intermediate
state between $S_4$ and $J_4$ is given by
\begin{equation}\label{e3.17}
(u_4,v_4)=\Big(u_{+},-v_{-}\cdot\frac{u_{+}-u_{-}-1}{u_{+}-u_{-}+1}\Big).
\end{equation}

Thus, the global solutions of the perturbed Riemann problem \eqref{e1.1}
and \eqref{e1.3} can be illustrated by Figure \ref{fig2}(a) 
for $u_{-}-2<u_{+}<u_{-}-1$
and Figure \ref{fig2}(b) for $u_{-}-1<u_{+}<u_{-}$, respectively. The proof is
complete.
\end{proof}


\noindent\textbf{Case 3.2: $R+J$ and $S+J$.}
In this case, we investigate the interaction between a rarefaction
wave followed by a contact discontinuity starting from
$(-\varepsilon,0)$ and a shock wave followed by a contact
discontinuity starting from $(\varepsilon,0)$ (see Figure \ref{fig3}). This case
happens if and only if $u_{-}<u_{m}$ and $u_{+}<u_{m}<u_{+}+1$, such
that we have $u_{-}-1<u_{+}$. The propagation speed of the contact
discontinuity $J_1$ is $\tau_1=2u_{m}+1$ and that of the shock
wave $S_1$ is $\sigma_1=u_{m}+u_{+}$. Thus, it is easy to see
that $J_1$ and $S_1$ intersect in finite time. The intersection
$(x_1,t_1)$ is also given by \eqref{e3.2} which is the same as that of
Case3.1.

Analogously, the new local Riemann problem will also be formulated
at $(x_1,t_1)$, which also gives rise to a new shock wave
$S_2$ and a new contact discontinuity $J_3$ after the time
$t_1$. For the propagation speed of the shock wave $S_2$ is
$\sigma_2=u_{m}+u_{+}$ and that of the wave front in the
rarefaction wave $R$ is $2u_{m}$, $S_2$ and $R$ will meet whose
first intersection $(x_2,t_2)$ is determined by
\begin{equation}\label{e3.18}
 \begin{gathered}
 x_2+\varepsilon=2u_{m}t_2, \\
 x_2-\varepsilon=(u_{m}+u_{+})t_2,
 \end{gathered}
\end{equation}
which means that
\begin{equation}\label{e3.19}
(x_2,t_2)=\Big(\frac{\varepsilon(3u_{m}+u_{+})}{u_{m}-u_{+}},
\frac{2\varepsilon}{u_{m}-u_{+}}\Big).
\end{equation}

The shock wave begins to enter the rarefaction wave fan after the
interaction of $S_2$ and $R$ happens. At the same time, the new
shock wave $S_3$ and contact discontinuity $J_4$ are generated
and propagate forwards. Here we use $\Gamma:x=x(t)$ to express the
curve of $S_3$ who has the changing state variables
$(u_4,v_4)$ on the left-hand side and $(u_{5},v_{5})$ on the
right-hand side. It follows from \eqref{e2.1} that
\begin{equation}\label{e3.20}
\begin{gathered}
 v_4\cdot e^{2u_4}=v_{-}\cdot e^{2u_{-}}, \\
 x+\varepsilon=2u_4t,
 \end{gathered}
\end{equation}
which enables us to have
\begin{equation}\label{e3.21}
(u_4,v_4)=\Big(\frac{x+\varepsilon}{2t},v_{-}\exp\Big(2u_{-}
-\frac{x+\varepsilon}{t}\Big)\Big).
\end{equation}
On the other hand, we always have $u_{5}=u_3=u_{+}$ for all the
characteristic lines in the rarefaction wave become the contact
discontinuities when across the shock wave $S_3$. In addition, if
the matched state $(u_{5},v_{5})$ can be connected to the
corresponding one $(u_4,v_4)$ by a shock wave, then they should
satisfy
\begin{equation}\label{e3.22}
\frac{v_{5}}{v_4}=\frac{u_4-u_{5}+1}{u_{5}-u_4+1}.
\end{equation}
Through a tedious and detailed calculation, one obtains
\begin{equation}\label{e3.23}
(u_{5},v_{5})
=\Big(u_{+},v_{-}\cdot\frac{x+\varepsilon-2t(u_{+}-1)}{2t(u_{+}+1)
-x-\varepsilon}\cdot
\exp\Big(2u_{-}-\frac{x+\varepsilon}{t}\Big)\Big).
\end{equation}
Thus, the curve of $S_3$ is determined by
\begin{equation}\label{e3.24}
\sigma_3(t)=\frac{dx}{dt}=u_{+}+\frac{x+\varepsilon}{2t},
\end{equation}
in which the initial condition $x(t_2)=x_2$ is given by \eqref{e3.19},
whose unique solution may be written as
\begin{equation}\label{e3.25}
x(t)=2u_{+}t+2\sqrt{2\varepsilon t(u_{m}-u_{+})}-\varepsilon,\quad\quad t\geq t_2.
\end{equation}
It follows from \eqref{e3.24} and \eqref{e3.25} that
\begin{equation}\label{e3.26}
\frac{d^{2}x}{dt^{2}}=-\frac{1}{2}\sqrt{\frac{2\varepsilon (u_{m}-u_{+})}{t^{3}}}<0,
\end{equation}
which means that $S_3$ begins to decelerate and is not a straight
line any more after the time $t_2$.

Taking into account the comparison of values between $u_{-}$ and
$u_{+}$, let us also use the following lemma to explain that if
$S_3$ is able to cancel the rarefaction wave $R$ completely or
not.

\begin{figure}[ht]
\begin{center}
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\end{center}
\caption{Interaction between $R+J$ and $S+J$ 
for two  situations where both $u_{-}<u_{m}$ and
$u_{+}<u_{m}<u_{+}+1$ should be satisfied.}
\label{fig3}
\end{figure}

\begin{lemma} \label{lem3.2}
If $u_{-}-1<u_{+}<u_{-}$, then the shock wave $S_3$ has the
ability to cancel the whole rarefaction wave $R$ completely in
finite time. Otherwise, if $u_+>u_-$, then the shock wave $S_3$
has no ability to penetrate
 the rarefaction wave $R$ completely in finite time and finally takes 
the line $x+\varepsilon=2u_{+}t$ as its asymptote.
\end{lemma}


\begin{proof}
If $u_{-}-1<u_{+}<u_{-}$, then the shock wave $S_3$ is continuous
to penetrate the rarefaction wave $R$ and is able to cancel the
whole rarefaction wave $R$ completely in finite time (see Figure \ref{fig3}(a)).
During the process of penetration, the local Riemann problem will be 
formulated on every point of the shock curve $S_3$
and new contact discontinuities will
be continuously produced along
 with the shock curve $S_3$. This is due to the fact that the shock curve 
$S_3$ has the varying left state $(u_4,v_4)$ supported on each characteristic 
line in the rarefaction wave fan $R$ which should be connected with the 
matched right state $(u_{5},v_{5})$ exactly by a shock wave followed by a 
contact discontinuity.
 It is clear to see that all  the contact discontinuities have the same 
propagation speed  $2u_{+}+1$, thus they are parallel to each other.
Finally, $S_3$ will meet the wave back in the rarefaction wave $R$
 at the intersection $(x_3,t_3)$, which can be calculated by
\begin{equation}\label{e3.27}
\begin{gathered}
 x_3+\varepsilon=2u_{-}t_3, \\
 x_3=2u_{+}t_3+2\sqrt{2\varepsilon  t_3(u_{m}-u_{+})}-\varepsilon.
 \end{gathered}
\end{equation}
Thus, we have
\begin{equation}\label{e3.28}
(x_3,t_3)=\Big(\frac{4\varepsilon u_{-}(u_{m}-u_{+})}{(u_{-}-u_{+})^2}
-\varepsilon,\frac{2\varepsilon(u_{m}-u_{+})}{(u_{-}-u_{+})^2}\Big).
\end{equation}

After the time $t_3$, the shock wave is denoted with $S_4$ whose
propagation speed is $\sigma_4=u_{-}+u_{+}$. The state
$(u_6,v_6)$ between $S_4$ and $J_{5}$ has the same
representation as that in \eqref{e3.17}. Consequently, a new shock wave
$S_4$ followed by a contact discontinuity $J_{5}$ will be
generated after the time $t_3$, in which the propagation speeds of
$S_4$ and $J_{5}$ are $\sigma_4=u_{-}+u_{+}$ and
$\tau_{5}=2u_{+}+1$ respectively.

Otherwise, if $u_+>u_-$, then the shock wave $S_3$ is continuous
to penetrate the rarefaction wave $R$ but unable to cancel the
whole
 $R$ completely in finite time (see Figure \ref{fig3}(b)).
During the process of penetration, it can be derived from \eqref{e3.24} and
\eqref{e3.25} that
\begin{equation}\label{e3.30}
\sigma_3(t)=\frac{dx}{dt}=2u_{+}+\sqrt{\frac{2\varepsilon(u_{m}-u_{+})}{t}}.
\end{equation}
Thus, it is shown that $\sigma_3(t)\to 2u_{+}$ as
$t\to\infty$ for given $\varepsilon >0$. Thus, when
$u_{+}>u_{-}$, the shock wave $S_3$ has the characteristic line
$x(t)=2u_{+}t-\varepsilon$ in the rarefaction wave fan $R$ as its
asymptote in the end.
\end{proof}


\noindent\textbf{Case 3.3: $S+J$ and $R+J$.}
Let us consider the situation that the shock wave $S_1$ plus the
contact discontinuity $J_1$ emanates from $(-\varepsilon,o)$ and
 the rarefaction wave $R_1$ plus another contact
discontinuity $J_2$ emits from $(\varepsilon,0)$ (see Fig.4). The
occurrence of this case depends on the conditions
$u_{-}-1<u_{m}<u_{-}$ and $u_{m}<u_{+}$, which implies that
$u_{-}-1<u_{+}$. The propagation speed of $J_1$ is
$\tau_1=2u_{m}+1$ and that of the wave back in the rarefaction
wave $R_1$ is $2u_{m}$. Thus, it is easy to see that $J_1$ and
$R_1$ meet at a time. The intersection $(x_1,t_1)$ is
determined by
\begin{equation}\label{e3.30b}
\begin{gathered}
 x_1+\varepsilon=(2u_{m}+1)t_1, \\
 x_1-\varepsilon=2u_{m}t_1,
 \end{gathered}
\end{equation}
which implies 
\begin{equation}\label{e3.31}
 (x_1,t_1)=\Big(\varepsilon(4u_{m}+1),2\varepsilon\Big).
\end{equation}

The contact discontinuity $J_1$ begins to enter the rarefaction
wave fan after the interaction between $J_1$ and $R_1$ occurs.
It can be derived directly from \eqref{e2.1} that the state variable
$(u_2,v_2)$ in $R_1$ can be determined by
\begin{equation}\label{e3.32}
\begin{gathered}
v_{m}\cdot e^{2u_{m}}=v_2\cdot e^{2u_2}, \\
 x-\varepsilon=2u_2t,
 \end{gathered}
\end{equation}
which implies 
\begin{equation}\label{e3.33}
(u_2,v_2)=\Big(\frac{x-\varepsilon}{2t},v_{m}\exp\Big(2u_{m}
-\frac{x-\varepsilon}{t}\Big)\Big).
\end{equation}

 It is remarkable that the values of $u$ on the both sides of the contact
discontinuity should be equal. In other words, the rarefaction wave
cannot change its direction when across $J_1$. Thus, the state
variable $(u_3,v_3)$ in $R_2$ can also be determined by
\begin{equation}\label{e3.34}
\begin{gathered}
v_1\cdot e^{2u_1}=v_3\cdot e^{2u_3}, \\
 x-\varepsilon=2u_3t,
 \end{gathered}
\end{equation}
such that 
\begin{equation}\label{e3.35}
(u_3,v_3)=\Big(\frac{x-\varepsilon}{2t},v_1\exp\Big(2u_1-\frac{x-\varepsilon}{t}
\Big)\Big),
\end{equation}
in which $(u_1,v_1)$ is given by \eqref{e3.4}. Therefore, the contact
discontinuity $J_1$ crosses the rarefaction wave $R_1$ with a
varying propagation speed, which is determined by
\begin{equation}\label{e3.36}
\begin{gathered}
\frac{dx}{dt}=2u_2+1,\\
x-\varepsilon=2u_2t,
 \end{gathered}
\end{equation}
together with the initial condition $x_1=x(t_1)$ given by \eqref{e3.31},
which enables us to obtain
\begin{equation}\label{e3.37}
x(t)=t\ln\frac{t}{2\varepsilon}+2u_{m}t+\varepsilon.
\end{equation}
It follows from \eqref{e3.36} and \eqref{e3.37} that
\begin{equation}\label{e3.38}
\frac{d^{2}x}{dt^{2}}=\frac{1}{t}>0,
\end{equation}
which means that $J_3$ begins to accelerate and is not a straight
line any more after the interaction between $J_1$ and $R_1$.

Furthermore, it is shown that the contact discontinuity $J_1$ has
the ability to penetrate the entire rarefaction wave $R_1$ fully in
finite time and the terminal point $(x_2,t_2)$ can be calculated by
\begin{equation}\label{e3.39}
\begin{gathered}
x_2=t_2\ln\frac{t_2}{2\varepsilon}+2u_{m}t_2+\varepsilon,\\
x_2-\varepsilon=2u_+t_2,
 \end{gathered}
\end{equation}
which implies 
\begin{equation}\label{e3.40}
 (x_2,t_2)=\Big(4\varepsilon u_+\exp(2u_{+}-2u_m)+\varepsilon,2\varepsilon
\exp(2u_+-2u_m)\Big).
\end{equation}
After the time $t_2$, the contact discontinuity is denoted with
$J_3$. The state between $J_2$ and $J_3$ is given by
\begin{equation}\label{e3.41}
(u_4,v_4)=(u_{+},v_{m}\exp(2u_m-2u_+)).
\end{equation}
The contact discontinuity $J_3$ is parallel to $J_2$ for the reason
that $u_4=u_+$. Similarly, the state $(u_5,v_5)$ between $R_2$ and
$J_3$ can be calculated by
\begin{equation}\label{e3.42}
(u_{5},v_{5})=(u_+,v_1\exp(2u_1-2u_+)).
\end{equation}

Now, let us consider the interaction between $S_1$ and $R_2$. The
propagation speed of $S_1$ is $\sigma_1=u_-+u_m$ and that of the
wave back in the rarefaction wave $R_2$ is still equal to $2u_m$.
Thus, it is easy to see that $S_1$ catches up with $R_2$ in finite
time and the interaction $(x_3,t_3)$ can be calculated by
\begin{equation}\label{e3.43}
\begin{gathered}
 x_3-\varepsilon=2u_{m}t_3, \\
 x_3+\varepsilon=(u_{-}+u_{m})t_3,
 \end{gathered}
\end{equation}
which implies 
\begin{equation}\label{e3.44}
(x_3,t_3)=\Big(\frac{\varepsilon(3u_{m}+u_{-})}{u_{-}
-u_{m}},\frac{2\varepsilon}{u_{-}-u_{m}}\Big).
\end{equation}

After the time $t_3$, the shock wave enters the region
 of the rarefaction wave fan $R_2$ and is denoted with $S_2$ during the process 
of penetration.
 It is noticed that the state on the right-hand side of the shock wave 
$S_2$ is $(u_3,v_3)$,
 in which $u_3$ varies from $u_{m}$ to $u_{+}$ for $u_1=u_{m}$ and
 $u_{5}=u_{+}$. To study the problem that the shock wave $S_2$
 penetrates the rarefaction wave $R_2$ is essential to study
 infinitely many local Riemann problems. There is still a shock wave
 followed by a contact discontinuity for a local Riemann problem
 provided that $u_{-}-1<u_3<u_{-}$. Thus, there are infinitely many contact
 discontinuities generated during the process of penetration. As before,
 the value $u_3$ does not change when these contact
 discontinuities pass through the rarefaction wave $R_2$.
Therefore, the curve of the shock wave $S_2$ can be determined by
\begin{equation}\label{e3.45}
\begin{gathered}
\frac{dx}{dt}=u_-+u_3,\\
x-\varepsilon=2u_3t,
 \end{gathered}
\end{equation}
and the initial condition $x_3=x(t_3)$ is given by \eqref{e3.44}, which has
a unique solution
\begin{equation}\label{e3.46}
x(t)=2u_{-}t-2\sqrt{2\varepsilon t(u_{-}-u_{m})}+\varepsilon.
\end{equation}
It follows from \eqref{e3.45} and \eqref{e3.46} that
\begin{equation}\label{e3.47}
\frac{d^{2}x}{dt^{2}}=\frac{1}{2}\sqrt{\frac{2\varepsilon
(u_{-}-u_{m})}{t^{3}}}>0,
\end{equation}
which means that $S_2$ also accelerates during the process of
penetration.

As in Lemma \ref{lem3.2}, according to the values $u_+$ and $u_{-}-1$, let
us also use the following lemma to explain that if $S_2$ is able
to cancel the rarefaction wave $R_2$ completely or not.


\begin{figure}[ht]
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\end{center}
\caption{Interactions between $S+J$ and $R+J$ 
for two different situations when both $u_{-}-1<u_{m}<u_-$ and
$u_{m}<u_{+}$ should be satisfied.}
\label{fig4}
\end{figure}



\begin{lemma} \label{lem3.3}
 If $u_{-}-1<u_{+}<u_{-}$, then the shock wave $S_2$
has the ability to cancel the whole rarefaction wave $R_2$
completely in finite time and a new shock wave followed by a new
 contact discontinuity will be generated at last. Otherwise, if
$u_+>u_-$, then the shock wave $S_2$ is unable to pass through the
whole $R_2$ completely and finally takes the line
$x-\varepsilon=2u_-t$ as its asymptote.
\end{lemma}


\begin{proof}
If $u_{-}-1<u_{+}<u_{-}$, then the shock wave $S_2$ is able to
penetrate the whole $R_2$ fully at $(x_4,t_4)$ (see Figure \ref{fig4}(a)), which can be calculated by
\begin{equation}\label{e3.49}
 \begin{gathered}
 x_4-\varepsilon=2u_{+}t_4, \\
 x_4=2u_{-}t_4-2\sqrt{2\varepsilon t_4(u_{-}-u_{m})}+\varepsilon,
 \end{gathered}
\end{equation}
namely
\begin{equation}\label{e3.50}
(x_4,t_4)=\Big(\frac{4\varepsilon
u_{+}(u_{-}-u_{m})}{(u_{-}-u_{+})^2}+\varepsilon,
\frac{2\varepsilon(u_{-}-u_{m})}{(u_{-}-u_{+})^2}\Big).
\end{equation}

After the time $t_4$, we denote the shock wave by $S_3$ whose
propagation speed is $\sigma_3=u_{-}+u_{+}$. Obviously, a new
contact discontinuity $J_4$ will be produced with the speed
$\tau_4=2u_{+}+1$. In addition, the state $(u_6,v_6)$ between
 $S_3$ and $J_4$ is given by
\begin{equation}\label{e3.51}
(u_6,v_6)=\Big(u_{+},-v_{-}\cdot\frac{u_{+}-u_{-}-1}{u_{+}-u_{-}+1}\Big).
\end{equation}
Otherwise, if $u_+>u_-$, then the shock wave $S_2$ is unable to
cancel the whole $R_2$ completely in finite time and ultimately
has $x(t)=2u_{-}t+\varepsilon$ as its asymptote (see Figure \ref{fig4}(b)).
\end{proof}

\noindent\textbf{Case 3.4: $R+J$ and $R+J$.}
In this case, we are concerned with the situation that both a
rarefaction wave followed by a contact discontinuity start from the
initial points $(-\varepsilon,0)$ and $(\varepsilon,0)$
respectively. The occurrence of this case depends on the condition
$u_{-}<u_{m}<u_{+}$. For this case, we need only to consider the
situation that a contact discontinuity penetrates a rarefaction
wave, which can be dealt with similarly to that for Case 3.3. The
details are omitted here.


\section{Interactions of delta shock waves with classical waves}

 It is known that the delta shock wave occurs in the Riemann solution
of \eqref{e1.1} and \eqref{e1.2} for some specific Riemann initial data. It is
interesting to investigate the interactions between the delta shock
wave with the other elementary waves, including shock wave,
rarefaction wave and contact discontinuity. As before, we
continue to study the perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3} but
we need to require that at least one delta shock wave generates at
 $(-\varepsilon,0)$ or $(\varepsilon,0)$. More
precisely, we need to consider five possibilities when the delta
shock wave is involved at the initial time, according to the
different combinations from $(-\varepsilon,0)$ and $(\varepsilon,0)$
as follows:
(1) $\delta S$ and $S+J$;\quad (2) $S+J$ and $\delta S$;\quad 
(3) $\delta S$ and $\delta S$;\quad
(4) $\delta S$ and $R+J$;\quad (5) $R+J$ and $\delta S$.

 In addition, it should be emphasized that
the interactions between the delta shock wave with the other
elementary waves for the system \eqref{e1.4} have been considered 
in \cite{M.Nedeljkov4} by using the
method of split delta function. In order for completeness and self-contained,
we utilize a somewhat different technique from that in \cite{M.Nedeljkov4} 
to study the perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3}
when the delta shock wave is involved at the initial moment. In addition, 
the systems \eqref{e1.1} and \eqref{e1.4} are derived from very different 
physical models and used to describe different physical phenomena.
Let us see \cite{B.T.Hayes,D.Tan1} for the detailed comparison and contrast 
between the two systems.
\smallskip



\noindent\textbf{Case 4.1: $\delta S$ and $S+J$.}
For this case, we draw our attention on the interaction between a
delta shock wave starting from $(-\varepsilon,0)$ and a shock wave
followed by a contact discontinuity starting from $(\varepsilon,0)$
(see Figure \ref{fig5}(a)). This requires that both $u_{m}\leq u_{-}-1$ and
 $u_{m}-1<u_{+}<u_{m}$ should be satisfied.
The propagation speed of the delta shock wave $\delta S_1$ is
$\sigma_{\delta_1}=u_{-}+u_{m}$ and that of the shock wave $S$ is
$\sigma=u_{m}+u_{+}$. Thus, $\delta S_1$ overtakes $S$ in finite
time whose intersection is given by
\begin{equation}\label{e4.1}
(x_1,t_1)=\Big(\frac{\varepsilon(u_{-}+2u_{m}+u_{+})}{u_{-}-u_{+}},
\frac{2\varepsilon}{u_{-}-u_{+}}\Big).
\end{equation}
In addition, the strength of delta shock wave before the time $t_1$ 
can be calculated by
\begin{equation}\label{e4.2}
\beta(t)=\Big((u_{-}-u_{m})(v_{-}+v_{m})-(v_{m}-v_{-})\Big)t \quad
\text{for } 0\leq t \leq t_1.
\end{equation}
Then, the generalized Riemann problem for the system \eqref{e1.1} with 
the delta type initial data is formulated at
$(x_1,t_1)$, in which the initial data are expressed as
\begin{equation}\label{e4.3}
 u|_{t=t_1}=\begin{cases}
 u_{-}, & x<x_1, \\
 u_1, & x>x_1,
\end{cases}
\qquad
 v|_{t=t_1}
=\beta(t_1) \delta_{(x_1,t_1)}
+\begin{cases}
 v_{-}, & x<x_1 \\
 v_1, & x>x_1,
\end{cases}
\end{equation}
where $(u_1,v_1)=(u_{+},-v_{m}\cdot\frac{u_{+}-u_{m}-1}{u_{+}-u_{m}+1})$
 can be calculated by the same formula as in \eqref{e3.5}.

For $u_1=u_{+}<u_{m}\leq u_{-}-1$, a new delta shock wave 
$\delta S_2$ will be generated after the interaction between 
$\delta S_1$ and $S$. The curve of $\delta S_2$ is determined by the equation
\begin{equation}\label{e4.4}
\sigma_{\delta_2}=\frac{dx}{dt}=u_{-}+u_1=u_{-}+u_{+}
\end{equation}
and the initial condition $x(t_1)=x_1$ given by \eqref{e4.1}, which enables us to get the unique exact solution
\begin{equation}\label{e4.5}
x(t)=(u_{-}+u_{+})\Big(t-\frac{2\varepsilon}{u_{-}-u_{+}}\Big)
+\frac{\varepsilon(u_{-}+2u_{m}+u_{+})}{u_{-}-u_{+}} \quad
\text{for } t\geq t_1.
\end{equation}
The propagation speed of $\delta S_2$ is $\sigma_{\delta_2}=u_{-}+u_{+}$ 
and that of the contact discontinuity $J$ is $\tau=2u_{+}+1$. 
Thus, it is easy to see that $\delta S_2$ intersects $J$ at a time. 
The intersection $(x_2,t_2)$ is determined by
\begin{equation}\label{e4.6}
\begin{gathered}
 x_2-x_1=(u_{-}+u_{+})(t_2-t_1), \\
 x_2-\varepsilon=(2u_{+}+1)t_2,
 \end{gathered}
\end{equation}
which means 
\begin{equation}\label{e4.7}
(x_2,t_2)=\Big(\frac{2\varepsilon(2u_{+}+1)(u_{-}-u_{m})}{(u_{-}-u_{+})(u_{-}-u_{+}-1)}
+\varepsilon,
\frac{2\varepsilon(u_{-}-u_{m})}{(u_{-}-u_{+})(u_{-}-u_{+}-1)}\Big).
\end{equation}
In addition, the
strength of delta shock wave between $t_1$ and $t_2$ can also be calculated by
\begin{equation}\label{e4.8}
\beta(t)=\beta(t_1)+\Big((u_{-}-u_{+})(v_{-}+v_1)-(v_1-v_{-})\Big)(t-t_1)
\quad \text{for } t_1< t \leq t_2.
\end{equation}

After the time $t_2$, the delta shock wave passes through $J$ with
the same propagation speed as before, but is at the different growth
rates for the strength of delta shock wave. This is due to the fact that 
the propagation speed of delta shock wave is calculated by 
$\sigma_{\delta}=u_{l}+u_{r}$ which is controlled completely by the state 
variable $u$, where $u_{l}$ stands for the state on the left-hand side of 
delta shock curve and $u_{r}$ expresses the state on the right-hand side 
of delta shock curve.
Taking into account $u_1=u_{+}$, we always have the states $u_{l}=u_{-}$ 
and $u_{r}=u_{+}$ on two sides of delta shock curve such that the propagation 
speed of delta shock wave keeps $u_{-}+u_{+}$ invariant after the time $t_1$. 
On the other hand, the strength of delta shock wave depends on both the state 
variables $u$ and $v$. Thus, the growth rate of delta shock wave also changes 
when across $J$ for the reason that the state variable $v$ changes from $v_1$ 
to $v_{+}$. After the time $t_2$, the strength of delta shock wave can be 
calculated by
\begin{equation}\label{e4.9}
\beta(t)=\beta(t_2)+\Big((u_{-}-u_{+})(v_{-}+v_{+})-(v_{+}-v_{-})\Big)(t-t_2)
\quad \text{for } t>t_2.
\end{equation}
\smallskip

\noindent\textbf{Case 4.2: $S+J$ and $\delta S$.}
In this case, we are concerned with the situation that a shock wave
followed by a contact discontinuity starts from $(-\varepsilon,0)$
and a delta shock wave emits from $(\varepsilon,0)$. This case
arises when both $u_{-}-1<u_{m}<u_{-}$ and $u_{+}\leq u_{m}-1$ occur.
This situation can be dealt with similarly to that for Case 4.1. 
Let us draw Figure \ref{fig5}(b) to illustrate this situation for
comparison and then give the detailed explanations for Figure \ref{fig5}(b) below.
The propagation speeds of $J$ and $\delta S_1$ are given by
 $\tau=2u_{m}+1$ and $\sigma_{\delta_1}=u_{m}+u_{+}$ respectively, 
such that they will meet at the intersection given by
\begin{equation}\label{e4.10}
(x_1,t_1)=\Big(\frac{\varepsilon(3u_{m}+1+u_{+})}{u_{m}+1-u_{+}},
\frac{2\varepsilon}{u_{m}+1-u_{+}}\Big).
\end{equation}

\begin{figure}[ht]
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\put(104.63,4.46){\makebox(0,0)[cc]{\footnotesize$-\varepsilon$}}
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}} \put(119.2,-0.31){\makebox(0,0)[cc]{(b) $S+J$ and $\delta S$}}
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S_2$}}}
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{\mbox{\tiny $-$}}}$}}%
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{\mbox{\tiny $m$}}}$}}%
\put(139.63,12.63){\makebox(0,0)[cc]{\footnotesize${\rlap{$\bigcirc$}\
{\mbox{\tiny $+$}}}$}}%
\put(114.88,20.88){\makebox(0,0)[cc]{\footnotesize${\rlap{$\bigcirc$}\
{\mbox{\tiny $1$}}}$}}%
\put(128.68,19.5){\makebox(0,0)[cc]{\footnotesize$(x_1,t_1)$}}
\put(118.13,30.08){\makebox(0,0)[cc]{\footnotesize$(x_2,t_2)$}}
\end{picture}
\end{center}
\caption{Interaction between $\delta S$ and $S+J$ when
$u_{m}\leq u_{-}-1$ and $u_{m}-1<u_{+}<u_{m}$ on the left, and 
interaction between $S+J$ and $\delta S$ when $u_{-}-1<u_{m}<u_{-}$
and $u_{+}\leq u_{m}-1$ on the right.}
\label{fig5}.
\end{figure}

The state $(u_1,v_1)$ between $S$ and $J$ can be calculated by the same 
formula as in \eqref{e3.4}. For $u_1=u_{m}$, the delta shock wave $\delta S_1$ 
cannot change its direction when across $J$.
Thus, the intersection of $S$ and $\delta S_1$ can be calculated by
\begin{equation}\label{e4.11}
\begin{gathered}
 x_2+\varepsilon=(u_{-}+u_{m})t_2, \\
 x_2-\varepsilon=(u_{m}+u_{+})t_2,
 \end{gathered}
\end{equation}
in which $u_{-}+u_{m}$ is the propagation speed of $S$, such that we have
\begin{equation}\label{e4.12}
(x_2,t_2)=\Big(\frac{\varepsilon(u_{-}+2u_{m}+u_{+})}{u_{-}-u_{+}},
\frac{2\varepsilon}{u_{-}-u_{+}}\Big).
\end{equation}
Consequently, the generalized Riemann problem for the system \eqref{e1.1} 
with the delta type initial data is also formulated at
$(x_2,t_2)$, in which the initial data are expressed as
\begin{equation}\label{e4.13}
 u|_{t=t_2}= \begin{cases}
 u_{-}, & x<x_2, \\
 u_{+}, & x>x_2,
\end{cases}
\qquad
 v|_{t=t_2}=\beta(t_2) \delta_{(x_2,t_2)}+
\begin{cases}
 v_{-}, & x<x_2 \\
 v_{+}, & x>x_2.
\end{cases}
\end{equation}
We can obtain $u_{+}<u_{-}-1$ from $u_{m}<u_{-}$ and $u_{+}\leq u_{m}-1$,
thus the interaction between $S$ and $\delta S_1$ generates a new delta 
shock wave denoted with $\delta S_2$
in Figure \ref{fig5}(b).

In addition, the strength of delta shock wave can be calculated respectively by
\begin{gather}\label{e4.14}
\beta(t)=\Big((u_{m}-u_{+})(v_{m}+v_{+})-(v_{+}-v_{m})\Big)t \quad
\text{for } 0\leq t \leq t_1, \\
\label{e4.15}
\beta(t)=\beta(t_1)+\Big((u_{m}-u_{+})(v_1+v_{+})-(v_{+}-v_1)\Big)(t-t_1)
\quad \text{for } t_1< t \leq t_2, \\
\label{e4.16}
\beta(t)=\beta(t_2)+\Big((u_{-}-u_{+})(v_{-}+v_{+})-(v_{+}-v_{-})\Big)(t-t_2)
\quad \text{for } t>t_2,
\end{gather}
in which $v_1=-v_{-}\cdot\frac{u_{m}-u_{-}-1}{u_{m}-u_{-}+1}$
is given by \eqref{e3.4}.
\smallskip

\noindent\textbf{Case 4.3: $\delta S$ and $\delta S$.}
In this case, we consider the interaction of two delta shock waves
starting from $(-\varepsilon,0)$ and $(\varepsilon,0)$ respectively.
This case happens if and only if $u_{m}\leq u_{-}-1$ and
$u_{+} \leq u_{m}-1$. Let us use $\delta S_1$ and $\delta S_2$ to denote 
the delta shock waves originating from the initial points
$(-\varepsilon,0)$ and $(\varepsilon,0)$ respectively. 
The propagation speed of $\delta S_1$ is $u_{-}+u_{m}$ and that of $\delta S_2$ 
is $u_{m}+u_{+}$,
and thus they will meet in finite time and the intersection of $\delta S_1$ 
and $\delta S_2$ can also be calculated by
the formula \eqref{e4.12}. Before the time $t_1=\frac{2\varepsilon}{u_{-}-u_{+}}$, 
the strengths of $\delta S_1$ and $\delta S_2$ can be calculated respectively by
\begin{gather}\label{e4.17}
\beta_1(t)=\Big((u_{-}-u_{m})(v_{-}+v_{m})-(v_{m}-v_{-})\Big)t, \\
\label{e4.18}
\beta_2(t)=\Big((u_{m}-u_{+})(v_{m}+v_{+})-(v_{+}-v_{m})\Big)t.
\end{gather}

At the point $(x_1,t_1)$, the delta-type initial data can also be formulated as
\begin{equation}\label{e4.19}
 u|_{t=t_1}=\begin{cases}
 u_{-}, & x<x_1, \\
 u_{+}, & x>x_1,
\end{cases}
\qquad
 v|_{t=t_1}= \beta(t_1) \delta_{(x_1,t_1)}+
\begin{cases}
 v_{-}, & x<x_1 \\
 v_{+}, & x>x_1,
\end{cases}
 \end{equation}
in which the strength $\beta(t_1)=\beta_1(t_1)+\beta_2(t_1)$ is the sum of 
the strengths of $\delta S_1$ and $\delta S_2$ at the point $(x_1,t_1)$
and thus can be calculated by
\begin{equation}\label{e4.20}
\beta(t_1)=\frac{2\varepsilon}{u_{-}-u_{+}}\cdot\Big((u_{-}-u_{m})(v_{-}+v_{m})
+(u_{m}-u_{+})(v_{m}+v_{+})-(v_{+}-v_{-})\Big).
\end{equation}
It can be obtained that $u_{+} \leq u_{-}-2$, thus
the wave interaction has a relatively simpler
structure for this case, namely two delta shock waves coalesce into
one delta shock wave when they meet. Let us use $\delta S_3$ to denote 
the new delta shock wave whose strength can be calculated by
\begin{equation}\label{e4.21}
\beta(t)=\beta(t_1)+\Big((u_{-}-u_{+})(v_{-}+v_{+})-(v_{+}-v_{-})\Big)(t-t_1)
\quad \text{for } t>t_1,
\end{equation}
in which $t_1=\frac{2\varepsilon}{u_{-}-u_{+}}$ and $\beta(t_1)$ is given 
by \eqref{e4.20}.
\smallskip

\noindent\textbf{Case 4.4: $\delta S$ and $R+J$.}
In this case, let us investigate the interaction between a delta
shock wave $\delta S_1$ emanating from $(-\varepsilon,0)$ and a
rarefaction wave $R_1$ followed by a contact discontinuity $J_1$
starting from $(\varepsilon,0)$ (see Fig.6). This is possible to
happen if and only if both $u_{m}\leq u_{-}-1$ and $u_{m}<u_{+}$ hold.
The propagation speed of the delta shock wave $\delta S_1$ is
$\sigma_{\delta_1}=u_{-}+u_{m}$ and the wave back in the
rarefaction wave propagates with the speed $2u_{m}$. Thus, 
$\delta S_1$ and $R$ intersect at the point
\begin{equation}\label{e4.22}
(x_1,t_1)=\Big(\frac{\varepsilon(3u_{m}+u_{-})}{u_{-}-u_{m}},
\frac{2\varepsilon}{u_{-}-u_{m}}\Big),
\end{equation}
and the strength of delta shock wave at $(x_1,t_1)$ is given by
\begin{equation}\label{e4.23}
\beta(t_1)=\Big((u_{-}-u_{m})(v_{-}+v_{m})-(v_{m}-v_{-})\Big)t_1.
\end{equation}

After the time $t_1$, the delta shock wave enters the rarefaction
wave fan $R_1$ where it is denoted with $\delta S_2$, whose
expression can be calculated by
\begin{equation}\label{e4.24}
\sigma_{\delta_2}(t)=\frac{dx}{dt}=u_{-}+\frac{x-\varepsilon}{2t},
\end{equation}
where the initial condition $x(t_1)=x_1$ is given by \eqref{e4.22},
which has a unique solution
\begin{equation}\label{e4.25}
x(t)=2u_{-}t-2\sqrt{2\varepsilon t(u_{-}-u_{m})}+\varepsilon, \quad 
\text{for } t\geq t_1.
\end{equation}
Thus, we have $\frac{d^{2}x}{dt^{2}}>0$, namely $\delta S_2$
begins to accelerate when it enters the rarefaction wave fan $R_1$. 
As before, it is easy to get that the state
 $(u_1,v_1)$ in $R_1$ and the state $(u_2,v_2)$ between $R_1$ and $J_1$
 are given respectively by
\begin{equation}\label{e4.26}
(u_1,v_1)=\Big(\frac{x-\varepsilon}{2t},v_{m}\exp(2u_{m}-\frac{x-\varepsilon}{t})\Big),
\end{equation}
\begin{equation}\label{e4
.27}
(u_2,v_2)=\Big(u_{+},v_{m}\exp(2u_{m}-2u_{+})\Big).
\end{equation}

In view of the different relations among the values $u_+$, $u_{-}$
and $u_{-}-1$, the discussions should be divided into three
different cases which can be fully depicted in the following lemma.

\begin{figure}[ht]
\begin{center}
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\caption{Interactions between $\delta S$ and $R+J$ 
for two different situations where both $u_{m}\leq u_{-}-1$
and $u_{m}<u_{+}$ should be satisfied.}
\label{fig6}
\end{figure}


\begin{lemma} \label{lem4.1}
If $u_{+}<u_{-}-1$, then the delta shock wave is able to cancel the
whole rarefaction wave $R_1$ completely in finite time. Otherwise,
if $u_{-}-1<u_{+}$, then the delta shock wave is unable to cancel
the whole $R_1$ completely in finite time and is divided into a
shock wave and a delta contact discontinuity when it passes through
the characteristic line with the state satisfying $u_1=u_{-}-1$.
\end{lemma}


\begin{proof}
If $u_{+}<u_{-}-1$, then $\delta S_2$ is able to cancel
 the whole $R_1$ at $(x_2,t_2)$ which can be calculated by
\begin{equation}\label{e4.28}
\begin{gathered}
 x_2-\varepsilon=2u_{+}t_2, \\
 x_2=2u_{-}t_3-2\sqrt{2\varepsilon t_2(u_{-}-u_{m})}+\varepsilon,
 \end{gathered}
\end{equation}
namely,
\begin{equation}\label{e4.29}
(x_2,t_2)=\Big(\frac{4\varepsilon
u_{+}(u_{-}-u_{m})}{(u_{-}-u_{+})^2}+\varepsilon,
\frac{2\varepsilon(u_{-}-u_{m})}{(u_{-}-u_{+})^2}\Big).
\end{equation}
The strength of $\delta S_2$ during the process of penetration can
be calculated by
\begin{equation}\label{e4.30}
\beta(t)=\beta(t_1)+\Big((u_{-}-u_1)(v_{-}+v_1)-(v_1-v_{-})\Big)(t-t_1),
\quad \text{for } t_1< t \leq t_2,
\end{equation}
in which $t_1$, $\beta(t_1)$ and $(u_1,v_1)$ are given by
\eqref{e4.22}, \eqref{e4.23} and \eqref{e4.26} respectively. 
After the time $t_2$, the situation is similar to that for Case 4.1. 
In other words, the delta
shock wave $\delta S_3$ propagates with the invariant speed
$u_{-}+u_{+}$, only the strength of $\delta S_3$ adds up at the
different rates when it passes through $J_1$.

Otherwise, if $u_{-}-1<u_{+}$, then the delta shock wave is unable
to cross the entire $R_1$ in finite time and should be decomposed
into a shock wave $S$ and a delta contact discontinuity $\delta J$
when it passes through the characteristic line with $u_1=u_{-}-1$
in the rarefaction wave fan $R_1$. This is due to the fact that
the inequality $u-1<u_{+}$ cannot always hold
 for the varying states $(u,v)$ supported on the characteristic lines in the rarefaction wave fan $R_1$.
More precisely, the critical
point for the decomposition of $\delta S_2$ into $S$ and $\delta
J$ can be calculated by
\begin{equation}\label{e4.31}
\begin{gathered}
 x_2-\varepsilon=2(u_{-}-1)t_2, \\
 x_2=2u_{-}t_2-2\sqrt{2\varepsilon t_2(u_{-}-u_{m})}+\varepsilon,
 \end{gathered}
\end{equation}
which enables us to have
\begin{equation}\label{e4.32}
(x_2,t_2)=\Big(4\varepsilon(u_{-}-u_{m})(u_{-}-1)
+\varepsilon,2\varepsilon(u_{-}-u_{m})\Big).
\end{equation}

The curve of the delta contact discontinuity $\delta J$ can be
calculated as
\begin{equation}\label{e4.33}
\tau_{\delta}(t)=\frac{dx}{dt}=\frac{x-\varepsilon}{t}+1,
\end{equation}
in which the initial condition $x(t_2)=x_2$ is given by \eqref{e4.32}.
Analogously, the expression of $\delta J$ can be given in the form
\begin{equation}\label{e4.34}
x(t)=t\Big(\ln t-\ln(2\varepsilon(u_{-}-u_{m}))+2u_{-}-2\Big)+\varepsilon,\quad
t\geq t_2.
\end{equation}
As in Case 3.3, the delta contact discontinuity $\delta J$ is able
to penetrate the whole $R_1$ in finite time and finally propagates
forwards with the invariant speed $2u_{+}+1$.

Let us turn our attention on the shock wave $S$. Actually, the shock
wave $S$ is the continuation of the delta shock wave $\delta S_2$.
Thus, the curve of the shock wave $S$ can also be expressed by
\eqref{e4.25}. Consequently, the shock wave $S$ continues to penetrate the
rarefaction wave $R_2$ and the situation is similar to that for
Case 3.3. For the reason that the propagation speed of $S$ depends 
entirely on the state variable $u$ on both sides of the shock curve, 
in which the left-hand state variable $u$ is always $u_{-}$
and the right-hand state variable $u$ supported on each characteristic 
line in the rarefaction wave fan remains unchanged when
the characteristic line passes through $\delta J$ and $J$.
 That is to say, we need also consider two subcases
$u_{-}-1<u_{+}<u_{-}$ and $u_{+}>u_{-}$, which can be illustrated by
Lemma \ref{lem3.3}.
\end{proof}


\noindent\textbf{Case 4.5: $R+J$ and $\delta S$.}
In the end, we study the situation when the rarefaction
wave $R$ followed by the contact discontinuity $J_1$ starts from
$(-\varepsilon,0)$ and the delta shock wave $\delta S_1$ starts
from $(\varepsilon,0)$ (see Fig.7). This case arises when both
$u_{-}<u_{m}$ and $u_{+} \leq u_{m}-1$ happen. The propagation speeds of
$J_1$ and $\delta S_1$
 are $\tau_1=2u_{m}+1$ and $\sigma_{\delta_1}=u_{m}+u_{+}$, respectively. 
Thus, $\delta S_1$ and $J_1$ meet in finite time whose intersection is 
\begin{equation}\label{e4.35}
 (x_1,t_1)=\Big(\frac{\varepsilon(3u_{m}+u_{+}+1)}{u_{m}-u_{+}+1},
\frac{2\varepsilon}{u_{m}-u_{+}+1}\Big).
\end{equation}
The strength of $\delta S_1$ at $(x_1,t_1)$ is 
\begin{equation}\label{e4.36}
\beta(t_1)=\Big((u_{m}-u_{+})(v_{m}+v_{+})-(v_{+}-v_{m})\Big)t_1.
\end{equation}

After the time $t_1$, the delta shock wave $\delta S_1$ passes
through $J_1$ with the same speed as before and consequently
enters the rarefaction wave $R$ from the point $(x_2,t_2)$ which
can be obtained as
\begin{equation}\label{e4.37}
(x_2,t_2)=\Big(\frac{\varepsilon(3u_{m}+u_{+})}{u_{m}-u_{+}},
\frac{2\varepsilon}{u_{m}-u_{+}}\Big).
\end{equation}
The strength of $\delta S_1$ at $(x_2,t_2)$ can be calculated as
\begin{equation}\label{e4.38}
\beta(t_2)=\beta(t_1)+\Big((u_1-u_{+})(v_1+v_{+})-(v_{+}-v_1)\Big)(t_2-t_1),
\end{equation}
in which
\begin{equation}\label{e4.39}
(u_1,v_1)=\Big(u_{m},v_{-}\exp(2u_{-}-2u_{m})\Big).
\end{equation}

After the time $t_2$, the delta shock wave enters the rarefaction
wave fan $R$ where it is denoted with $\delta S_2$. As
before, the curve of $\delta S_2$ is determined by
\begin{equation}\label{e4.40}
\sigma_{\delta_2}(t)=\frac{dx}{dt}=u_{+}+\frac{x+\varepsilon}{2t},
\end{equation}
in which the initial condition $x(t_2)=x_2$ is given by \eqref{e4.37}.
It is easy to obtain a unique solution
\begin{equation}\label{e4.41}
x(t)=2u_{+}t+2\sqrt{2\varepsilon
t(u_{m}-u_{+})}-\varepsilon,\quad\quad t\geq t_2.
\end{equation}
It follows from \eqref{e4.40} and \eqref{e4.41} that $\frac{d^{2}x}{dt^{2}}<0$,
which means that $\delta S_2$ begins to decelerate and is not
 a straight line any more.

 The state $(u_2,v_2)$ in the rarefaction wave $R$ can be
obtained as
\begin{equation}\label{e4.42}
(u_2,v_2)=\Big(\frac{x+\varepsilon}{2t},v_{m}\exp\Big(2u_{-}
-\frac{x+\varepsilon}{t}\Big)\Big).
\end{equation}
As in Lemma \ref{lem4.1}, one can see that if $u_{-}>u_{+}+1$, then the
delta shock wave is able to cancel the whole rarefaction wave $R$
completely in finite time and consequently propagates with the
invariant speed $u_{-}+u_{+}$. Otherwise, if $u_{-}<u_{+}+1$, then
the delta shock wave cannot cross the whole rarefaction wave $R$
completely in finite time and may be also divided into a shock wave
and a delta contact discontinuity when it passes through the
characteristic line with the state satisfying $u_2=u_{+}+1$ in the 
rarefaction wave fan $R$.

\begin{figure}[ht]
\begin{center}
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\end{picture}
\end{center}
\caption{Interactions between $R+J$ and $\delta S$ 
for two different situations where both $u_{-}<u_{m}$ and
$u_{+} \leq u_{m}-1$ should be satisfied.}
\label{fig7}
\end{figure}

Analogously, the critical point for the decomposition of 
$\delta S_2$ into $S$ and $\delta J$ can be calculated by
\begin{equation}\label{e4.43}
\begin{gathered}
 x_3+\varepsilon=2(u_{+}+1)t_3, \\
 x_3=2u_{+}t_3+2\sqrt{2\varepsilon
t_3(u_{m}-u_{+})}-\varepsilon,
 \end{gathered}
\end{equation}
which enables us to obtain
\begin{equation}\label{e4.44}
(x_3,t_3)=\Big(4\varepsilon(u_{m}-u_{+})(u_{+}+1)-\varepsilon,
2\varepsilon(u_{m}-u_{+})\Big).
\end{equation}
Consequently, the curve of $\delta J$ can be expressed as
 \begin{equation}\label{e4.45}
 x(t)=(2u_{+}+1)(t-2\varepsilon(u_{m}-u_{+}))
+4\varepsilon(u_{m}-u_{+})(u_{+}+1)-\varepsilon.
 \end{equation}
On the other hand, the shock wave $S$ is still the continuation of
the delta shock wave $\delta S_2$ and can be expressed by \eqref{e4.41}.
Consequently, the shock wave $S$ continues to penetrate the
rarefaction wave $R$. If $u_{+}<u_{-}<u_{+}+1$, then the shock wave
$S$ is able to penetrate the whole $R$ completely in finite time.
Otherwise, if $u_-<u_+$, then the shock wave $S$ cannot penetrate
 $R$ completely in finite time and finally has the line $x+\varepsilon=2u_{+}t$ 
as its asymptote.

\section{Discussions and conclusions}

So far, the wave interaction problems for the system \eqref{e1.1} have been
investigated in detail. The global solutions of the perturbed
Riemann problem \eqref{e1.1} and \eqref{e1.3} are constructed fully for all the
situations. Now, we are in a position to consider whether the limits
of the solutions of the perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3}
are the corresponding ones of the Riemann problem \eqref{e1.1} and \eqref{e1.2} or
not as $\varepsilon \to 0$. Let us take Case 4.4 as an
example to explain our problem in detail.


 With $u_{m} \leq u_{-}-1$ and $u_{m}<u_{+}$ in mind for Case 4.4, 
we first consider the situation that
$u_{+} \leq u_{-}-1$. It is easy to see that all the points
$(-\varepsilon,0)$, $(\varepsilon,0)$, $(x_1,t_1)$,
$(x_2,t_2)$ and $(x_3,t_3)$ tend to the origin $(0,0)$ and
coincide with each other, such that there is only a delta shock wave
with the propagation speed $u_{-}+u_{+}$ in the limit situation. If
$u_{-}-1<u_{+}<u_{-}$, then the shock wave is able to cancel the
whole rarefaction wave completely. It can be seen from \eqref{e4.23} and
\eqref{e4.30} that $\beta(t_2)$ tends to zero for the reason that $t_2$
tends to zero as $\varepsilon\to0$, which implies that the
delta shock wave disappears and the delta contact discontinuity
becomes the contact discontinuity in the limit situation.
Furthermore, we can see that all the contact discontinuities
coincide with each other in the limit situation for they have the
same propagation speed and start from the origin $(0,0)$. Thus,
there is still $S+J$ for $u_{-}-1<u_{+}<u_{-}$ in the limit
situation. Otherwise, if $u_{+}>u_{-}$, then the limit situation is
similar and the difference only lies in that the shock wave is
unable to cancel the rarefaction wave completely. Thus, there is
$R+J$ for $u_{+}>u_{-}$ in the limit situation. Gathering the above
results together, we can see that the limits of the solutions of the
perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3} are identical with the
corresponding ones of the Riemann problem \eqref{e1.1} and \eqref{e1.2} as
$\varepsilon \to 0$ for Case 4.4.



The above method can also be generalized to the other cases and one
can discover that the large-time asymptotic solutions of the
perturbed Riemann problem \eqref{e1.1} and \eqref{e1.3} indeed coincide with the
corresponding ones of the Riemann problem \eqref{e1.1} and \eqref{e1.2}. That is
to say, the large-time asymptotic solutions of the perturbed Riemann
problem \eqref{e1.1} and \eqref{e1.3} is the delta shock wave for
$u_{+} \leq u_{-}-1$, the shock wave followed by the contact
discontinuity for $u_{-}-1<u_{+}<u_{-}$ and the rarefaction wave
followed by the contact discontinuity for $u_{+}>u_{-}$.
Let us call that the solutions of the Riemann problem \eqref{e1.1} and \eqref{e1.2} are stable with respect to the
specific small perturbations \eqref{e1.3} of the Riemann initial data \eqref{e1.2} provided that the solutions of
the perturbed Riemann problem \eqref{e1.1}
and \eqref{e1.3} converge to the ones of the corresponding Riemann
problem \eqref{e1.1} and \eqref{e1.2} as $\varepsilon\to0$ in the sense of distributions in all kinds of situations.
In a word,
we can summarize our results in the following theorem.

\begin{theorem} \label{thm5.1}
The limits of the solutions to the perturbed Riemann problem \eqref{e1.1}
and \eqref{e1.3} are identical with the corresponding ones to the Riemann
problem \eqref{e1.1} and \eqref{e1.2} as $\varepsilon \to 0$ for all kinds
of situations. Thus, the conclusion can be drawn that the solutions
to the Riemann problem \eqref{e1.1} and \eqref{e1.2} are stable with respect to
such a local small perturbation \eqref{e1.3} of the Riemann initial data
\eqref{e1.2}.
\end{theorem}


\subsection*{Acknowledgements}

The authors would like to thank the anonymous referees for their useful comments 
on the original manuscript.
This work is partially supported by the Shandong
Provincial Natural Science Foundation (ZR2014AM024) and  the
National Natural Science Foundation of China (11441002).


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\end{document}