\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 126, pp. 1--16.\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-2016/126\hfil Stability of Riemann solutions]
{Structural stability of solutions to the Riemann problem for a
 non-strictly hyperbolic system with flux approximation}

\author[M. Sun \hfil EJDE-2016/126\hfilneg]
{Meina Sun}

\address{Meina Sun \newline
School of Mathematics and  Statistics Science,
Ludong University, Yantai 264025, China}
\email{smnwhy0350@163.com}

\thanks{Submitted December 9, 2015. Published May 19, 2016.}
\subjclass[2010]{35L65, 35L67, 35B30}
\keywords{Delta shock wave;  Riemann problem; non-strictly hyperbolic system;
\hfill\break\indent triangular system; flux approximation}

\begin{abstract}
 We study the Riemann problem for a non-strictly hyperbolic system of
 conservation laws under the linear approximations of flux functions with
 three  parameters. The approximated system also belongs to the type of
 triangular systems of conservation laws and this approximation does
 not change the structure of Riemann solutions to the original system.
 Furthermore, it is proven that the Riemann solutions to the
 approximated system converge to the corresponding ones to the original
 system as the perturbation parameter tends to zero.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Non-strictly hyperbolic systems of conservation laws have not only
important physical background but also special interest and
difficulty in mathematics. It is well known that the Cauchy problem
usually does not have a weak $L^{\infty}$-solution for some
non-strictly hyperbolic systems of conservation laws. Thus, the
measure-valued solution should be introduced into this nonclassical
situation, such as delta shock wave \cite{G.Q.Chen,W.Sheng1,D.Tan}
and singular shock wave \cite{B.L.Keyfitz,M.Nedeljkov}, which can
often provide a reasonable explanation for some physical phenomena.
However, the mechanism for the formation of delta shock wave cannot
be fully understood, though the necessity of delta shock wave is
obvious for the solutions of Riemann problems for some non-strictly
hyperbolic systems of conservation laws.

In this article, we are concerned with the  non-strictly
hyperbolic system of conservation laws
\begin{equation}\label{e1.1}
\begin{gathered}
u_t+(u^{2})_x=0, \\
v_t+(uv)_x=0.
\end{gathered}
\end{equation}
The system \eqref{e1.1} can be derived in \cite{D.Tan} directly from the
system of Euler gas dynamics by letting both the density and the
pressure to be constants in the momentum equation. The system \eqref{e1.1}
arises in several fields which can be used to model conservation
laws for some specific situations, such as magnetohydrodynamics,
elasticity and oil recovery process \cite{V.M.Shelkovich,H.Swan}.
The first equation in \eqref{e1.1} is just the inviscid Burgers equation
and the solutions of the Riemann problem are the classical entropy
solutions.
The  Dirac delta function is introduced as a part for $v$ in the second
equation in \eqref{e1.1} when the characteristic velocity $u$
 is discontinuous. In 1994, Tan, Zhang and Zheng \cite{D.Tan}
considered the Riemann problem for  \eqref{e1.1} and they
discovered that the form of the standard Dirac delta function
supported on a shock wave was used as a part in their Riemann
solutions for some specific initial data.  Since then, the delta
shock wave solution for  \eqref{e1.1} has been widely
investigated such as in \cite{C.Shen2,C.Shen3,A.Yao}.

The formation of delta shock wave has been extensively studied by
using the vanishing pressure approximation for the systems of
pressureless gas dynamics
\cite{G.Q.Chen,G.Q.Chen2,R.J.Leveque,J.Li1,D.Mitrovic,C.Shen1,G.Yin,G.Yin2}
and Chaplygin gas dynamics \cite{H.Cheng,W.Sheng1,H.Yang}, which is
a particular case of flux function approximation. Recently, the flux
function approximation with two parameters \cite{C.Shen} and three
parameters \cite{H.Yang1} has also been carried out for the systems
of pressureless gas dynamics. In the present paper, we consider the
linear approximations of flux functions in  \eqref{e1.1} as
follows:
\begin{equation}\label{e1.2}
\begin{gathered}
u_t+(u^{2}+\varepsilon \alpha u)_x=0, \\
v_t+(uv+\varepsilon \beta u+\varepsilon \gamma v)_x=0,
\end{gathered}
\end{equation}
where $\alpha,\beta,\gamma$ are arbitrary real constant numbers and
$\varepsilon$ is a sufficiently small positive number.
 More precisely, we are only concerned with the Riemann problem here,
 which is a special Cauchy problem with initial data
\begin{equation}\label{e1.3}
(u,v)(x,0)=\begin{cases}
(u_{-},v_{-}), & x<0,\\
(u_{+},v_{+}), & x>0,
 \end{cases}
\end{equation}
where $u_{\pm}$ and  $v_{\pm}$ are all given constants.

It is remarkable that  system \eqref{e1.1} is a particular example for
the triangular systems of conservation laws due to the special
structure where the evolution of the unknown variable $u$ does not
depend on the succeeding unknown variable $v$. It can be seen that
 system \eqref{e1.2} also belongs to the type of triangular systems of
conservation laws under the triangular linear approximations of flux
functions. It can be discovered that the delta shock wave  also
appears in the solution of the Riemann problem \eqref{e1.2} and \eqref{e1.3} for
some specific initial data. Furthermore, it is proven rigorously
that the limits of solutions to the Riemann problem \eqref{e1.2} and \eqref{e1.3}
converge to the corresponding ones of  the Riemann problem \eqref{e1.1} and
\eqref{e1.3} when the perturbation parameter $\varepsilon$ tends to zero.
In other words, the Riemann solutions of \eqref{e1.1} and \eqref{e1.3} is stable
with respect to the triangular linear approximations of flux
functions in the form of \eqref{e1.2}. Actually, one can see that the
Riemann solutions of \eqref{e1.1} and \eqref{e1.3} just translate in the $(x,t)$
plane under the triangular linear approximations of flux functions
in the form of \eqref{e1.2}. Thus, this triangular linear approximations of
flux functions in the form of \eqref{e1.2} does not change the structure of
 solutions to the Riemann problem \eqref{e1.1} and \eqref{e1.3}.

 In fact, the concept of Dirac delta function was first introduced into the
classical weak solution of hyperbolic conservation laws by
Korchinski \cite{D.J.Korchinski} in 1975 when he considered the
Riemann problem for the system
\begin{equation}\label{e1.4}
\begin{gathered}
u_t+(\frac{1}{2}u^{2})_x=0, \\
v_t+(\frac{1}{2}uv)_x=0,
\end{gathered}
\end{equation}
which has the trivial difference $u \to 2u$ from
\eqref{e1.1}. Since 1994, there are numerous excellent papers about the
concept of delta shock wave for the related equations and results,
see \cite{V.G.Danilov1,F.Huang,B.L.Keyfitz,M.Nedeljkov,W.Sheng} for
instance. At present, the
 vanishing pressure approach is one of the popular approaches
to study the formation of delta shock wave appearing in the Riemann
solution for some hyperbolic systems of conservation laws. In the
present paper, we consider the linear approximations of flux
functions with three parameters in the form of \eqref{e1.2} for a
particular triangular hyperbolic system of conservation laws which
has not been paid attention before.


This article is organized as follows. In section 2, we
describe simply the solutions of the
 Riemann problem \eqref{e1.1} and \eqref{e1.3} for completeness.
In section  3, the Riemann problem for the approximated system \eqref{e1.2}
is considered  and the Riemann solutions are constructed completely for six
different cases. In section 4, the limit of Riemann
solutions to the approximated system \eqref{e1.2} is taken by letting the
perturbation parameter $\varepsilon$ tend to zero, which is
identical with the corresponding ones to the original system.
Finally, the conclusion and discussion are drawn in section 5.

\section{Preliminaries}

In this section, we simply describe the results on the Riemann
problem \eqref{e1.1} and \eqref{e1.3}, which can be seen such as in \cite{D.Tan}.
The eigenvalues of  \eqref{e1.1} are $\lambda_1=u$ and
 $\lambda_2=2u$ and the corresponding
 right eigenvectors are $\overrightarrow{r}_1=(0,1)^{T}$ and
$\overrightarrow{r}_2=(1,v/u)^{T}$, respectively.
 It is noted that $\lambda_1<\lambda_2$ for $u>0$ and
$\lambda_1>\lambda_2$ for  $u<0$ here.
 Thus, \eqref{e1.1} is a non-strictly hyperbolic system.
It can be obtained directly that the characteristic field for $\lambda_1$
 is linearly degenerate and the characteristic field for $\lambda_2$
is genuinely nonlinear.

 Besides the constant state, it can be seen from  \cite{D.Tan} that the
self-similar waves $(u,v)(\xi)$ $(\xi=x/t)$ of the first family are
contact discontinuities denoted by $J$ as
 $$
 J:\xi=u_l=u_r,
 $$
and those of the second family are rarefaction waves denoted by $R$
as
$$
R:\xi=2u,\quad \frac{u}{v}=\frac{u_l}{v_l},\quad u_l<u_r,
$$
or shock waves denoted by $S$ as
$$
S:\xi=u_l+u_r,\quad \frac{u_r}{v_r}=\frac{u_l}{v_l},\quad
u_l>u_r>0,\quad \text{or } 0>u_l>u_r,
$$
in which the indices $l$ and $r$ stand for the left and right states
respectively. All the waves $J$, $R$ and $S$ are called as classical
waves here.

 For the case $u_{+}\leq 0 \leq u_{-}$, a solution
containing a weighted delta measure supported on a line should be
constructed. In order to define the delta shock wave solution to the
Riemann problem \eqref{e1.1} and \eqref{e1.3}, let us introduce the following
definitions below.


\begin{definition} \label{def2.1} \rm
 To define the measure solutions, the two-dimensional weighted delta measure
$w(s)\delta_{\Gamma}$ supported on a smooth curve
$\Gamma=\{(x(s),t(s)):a<s<b\}$ is defined by
\begin{equation}\label{e2.1}
\langle w(s)\delta_{\Gamma},\psi(x,t)\rangle
=\int^{b}_{a}w(s)\psi(x(s),t(s))ds,
\end{equation}
for any test function $\psi(x,t) \in C^{\infty}_0(R \times R_{+})$.
 \end{definition}

Now, let us introduce the definition of delta shock wave solution in
the framework introduced by Danilov and Shelkovich
\cite{V.G.Danilov2,V.G.Danilov3} and developed by Kalisch and
Mitrovic \cite{H.Kalisch,H.Kalisch2} below. Suppose that
$\Gamma=\{\gamma_{i}|i\in I\}$ is a graph in the closed
   upper half-plane $\{(x,t)|(x,t)\in (-\infty,\infty)\times
   [0,\infty)\}$ which contains Lipschitz continuous arcs $\gamma_{i}$ with $i\in
   I$ in which $I$ is a finite index set.  Suppose that
    $I_0$ is a subset of $I$ which contains all indices of arcs linking
to the $x$-axis and $\Gamma_0=\{x_{k}^{0}|k\in I_0\}$
     is the set of initial points of $\gamma_{k}$ with $k\in I_0$.


\begin{definition} \label{def2.2} \rm
Let $(u,v)$ be a pair of distributions where $v$ has the form
\begin{equation}  \label{e2.2}
v(x,t)=\hat{v}(x,t)+\alpha(x,t)\delta(\Gamma),
\end{equation}
in which $u,\hat{v}\in L^{\infty}(R\times R_{+})$ and the singular
part is defined by
\begin{equation}  \label{e2.3}
\alpha(x,t)\delta(\Gamma)=\sum_{i\in
I}\alpha_{i}(x,t)\delta(\gamma_{i}).
\end{equation}
Let us consider the delta shock wave type initial data
\begin{equation}  \label{e2.4}
(u,v)(x,0)=\Big(u_0(x),\hat{v}_0(x)+\sum_{k\in I_0}
\alpha_{k}(x^{0}_{k},0)\delta(x-x^{0}_{k})\Big),
\end{equation}
in which $u_0,\hat{v}_0\in L^{\infty}(R)$, then the pair of
distributions  $(u,v)$ are called as a generalized delta shock wave
solution for  \eqref{e1.1} with the delta shock wave type initial
data \eqref{e2.4} if the following integral identities
\begin{gather}  \label{e2.5}
 \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, \\
  \label{e2.6}
\begin{aligned}
&\int_{R_{+}}\int_{R}\left( \hat{v}\psi _t+u\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{gather}
hold for all test functions $\psi\in C_{c}^{\infty}(R\times R_{+})$,
in which $\frac{\partial \psi(x,t) }{\partial l}$ stands for the
tangential derivative of $\psi$ on the graph $\gamma_{i}$ and
$\int_{\gamma_{i}}$ expresses the line integral along
$\gamma_{i}$.
\end{definition}

With the above definition, a piecewise smooth solution of \eqref{e1.1}
and \eqref{e1.3} can be constructed for
the case  $u_{+}\leq 0 \leq u_{-}$ in the form
\begin{equation}\label{e2.7}
u(x,t)= \begin{cases}
   u_{-}, & x<\sigma_{\delta} t, \\
   u_{+}, & x>\sigma_{\delta} t,
 \end{cases}
 \quad  v(x,t)= \begin{cases}
    v_{-}, & x<\sigma_{\delta} t \\
    v_{+}, & x>\sigma_{\delta} t
 \end{cases} +w(t) \delta(x-\sigma_{\delta} t)
\end{equation}
where
\begin{equation}\label{e2.8}
  \sigma_{\delta}=u_{-}+u_{+}, \quad
  w(t)=(u_{-}v_{+}-u_{+}v_{-})t.
\end{equation}
The functions $w(t)$ and $\sigma_{\delta}$ express the strength and
propagation speed of delta shock wave, respectively.

The delta shock wave solution \eqref{e2.7} with \eqref{e2.8} satisfies the
 generalized Rankine-Hugoniot condition
\begin{equation}\label{e2.9}
\begin{gathered}
\frac{dx}{dt}=\sigma_{\delta}, \\
\frac{dw(t)}{dt}=\sigma_{\delta}[v]-[uv], \\
[u^{2}]=\sigma_{\delta}[u],
\end{gathered}
\end{equation}
where $[u]=u(x(t)+0,t)-u(x(t)-0,t)$, etc. In order to ensure the
uniqueness, the entropy condition of delta shock wave should be
proposed as $\lambda_{2r}\leq \lambda_{1r}\leq \sigma_{\delta} \leq
\lambda_{1l}\leq \lambda_{2l}$. It is an over-compressive condition
which implies that all the characteristics on both sides of the
delta shock wave curve are incoming.

The above constructed delta shock wave solution \eqref{e2.7} with \eqref{e2.8}
should satisfy
\begin{equation}\label{e2.10}
\begin{gathered}
\langle u, \psi_t \rangle+\langle u^{2}, \psi_x \rangle=0, \\
\langle v, \psi_t \rangle+\langle uv, \psi_x \rangle=0,
\end{gathered}
\end{equation}
for any test function $\psi(x,t) \in C^{\infty}_0(R \times
R_{+})$. In the above formula \eqref{e2.10},
 as in \cite{G.Q.Chen,W.Sheng,D.Tan}, we have
\begin{equation}\label{e2.11}
\langle v , \psi \rangle=\int^{\infty}_0 \int^{\infty}_{-\infty}
 v_0\psi \,dx\,dt+\langle w(t)\delta_S, \psi\rangle,
\end{equation}
\begin{equation}\label{e2.12}
\langle uv , \psi \rangle=\int^{\infty}_0 \int^{\infty}_{-\infty}
u_0 v_0\psi \,dx\,dt+\langle u_{\delta} w(t)\delta_S,
\psi\rangle,
\end{equation}
in which $u_0=u_{-}+[u]H(x-\sigma_{\delta} t)$,
$v_0=v_{-}+[v]H(x-\sigma_{\delta} t)$ and $u_0
v_0=u_{-} v_{-}+[uv]H(x-\sigma_{\delta} t)$. In order to require the solution
\eqref{e2.7} with \eqref{e2.8} to
satisfy \eqref{e2.10} in the sense of distributions, it is necessary to specify
the value of velocity $u$ along the trajectory of singularity.
Thus, $u_{\delta}$ is introduced in the formula \eqref{e2.12} which stands for
the assignment of $u$ on this delta shock wave curve $x=\sigma_{\delta} t$,
although the reasonable physical explanation may not be given clearly.
Then, the solution \eqref{e2.7} can be rewritten as
\begin{equation}\label{e2.13}
(u,v)(x,t)= \begin{cases}
   (u_{-}, v_{-}), & x<\sigma_{\delta} t, \\
   (u_{\delta},w(t) \delta(x-\sigma_{\delta} t)),   & x=\sigma_{\delta} t,  \\
   (u_{+}, v_{+}),     &  x>\sigma_{\delta} t,
 \end{cases}
\end{equation}
where $u_{\delta}=\sigma_{\delta}=u_{-}+u_{+}$. In fact, it can be seen
from \cite{D.Tan} that the delta shock wave solution \eqref{e2.13}
 with \eqref{e2.8} indeed satisfies \eqref{e2.10} in the sense of distributions.

With the entropy conditions of shock wave and delta shock wave
above, there exist six different configurations of solutions to the
Riemann problem \eqref{e1.1} and \eqref{e1.3} according to the values of $u_-$ and
$u_+$ as follows:
\begin{gather*}
\delta S(u_{+}\leq 0 \leq u_{-}), \quad
J+\overrightarrow{S}(0<u_+<u_-), \quad J+\overrightarrow{R}(0\leq u_-<u_+), \\
\overleftarrow{R}+\overrightarrow{R}(u_-<0<u_+), \quad
\overleftarrow{R}+J(u_-<u_+\leq0), \quad
\overleftarrow{S}+J(u_+<u_-<0).
\end{gather*}

\section{Riemann problems \eqref{e1.2} and \eqref{e1.3}}

In this section, we consider  \eqref{e1.2} and \eqref{e1.3}
for any given sufficiently small parameter $\varepsilon>0$.
System \eqref{e1.2} can be rewritten in the quasi-linear form
\begin{equation}\label{e3.1}
 \begin{pmatrix} u \\ v  \end{pmatrix}_t
+ \begin{pmatrix} 2u+\varepsilon \alpha  & 0 \\
   v+\varepsilon\beta & u+\varepsilon\gamma
   \end{pmatrix}
 \begin{pmatrix} u \\ v  \end{pmatrix}_{x}
=  \begin{pmatrix} 0 \\
  0
   \end{pmatrix}.
\end{equation}
It can be derived directly from \eqref{e3.1} that the two eigenvalues of
 system \eqref{e1.2} are
\begin{equation}\label{e3.2}
\lambda_1(u,v)=u+\epsilon \gamma, \quad
\lambda_2(u,v)=2u+\varepsilon\alpha.
\end{equation}
 It is clear  that \eqref{e1.2} is non-strictly hyperbolic
 for the reason that $\lambda_1<\lambda_2$ when $u>\varepsilon(\gamma-\alpha)$
 and $\lambda_1>\lambda_2$ when
$u<\varepsilon(\gamma-\alpha)$. The corresponding right eigenvectors
of $\lambda_{i}$ $(i=1,2)$ can be expressed respectively as
\begin{equation}\label{e3.3}
\overrightarrow{r}_1=(0,1)^{T}, \quad
\overrightarrow{r}_2=(u+\varepsilon\alpha-\varepsilon\gamma,v+\varepsilon\beta)^{T}.
\end{equation}
 By a direct calculation, we have $\nabla \lambda_1 \cdot
\overrightarrow{r}_1=0 $ and $\nabla \lambda_2 \cdot
\overrightarrow{r}_2=2(u+\varepsilon\alpha-\varepsilon\gamma)$, in
which $\nabla$ denotes the gradient with respect to $(u,v)$. Thus,
it can be concluded that the characteristic field for $\lambda_1$
is always linearly degenerate and the associated wave is the contact
discontinuity denoted by $J$ and  the characteristic field for
$\lambda_2$ is genuinely nonlinear provided that
$u\neq\varepsilon(\gamma-\alpha)$ and the associated wave is the
shock wave denoted by $S$ or the rarefaction wave denoted by $R$.
The Riemann invariants along the characteristic fields are
\begin{equation}\label{e3.4}
w=u, \quad
z=\frac{v+\varepsilon\beta}{u+\varepsilon\alpha-\varepsilon\gamma}.
 \end{equation}


Since both  system \eqref{e1.2} and the Riemann initial data \eqref{e1.3} are
unchanged under the coordinate transformations in the $(x,t)$ plane:
$(x,t)\to (kx,kt)$ where $k$ is a constant, we need to look for the
self-similar solutions of the form
  \begin{equation}\label{e3.5}
  (u,v)(x,t)=(u,v)(\xi),   \quad \xi=x/t.
  \end{equation}
Hence, the Riemann problem \eqref{e1.2} and \eqref{e1.3} is reduced to the
 boundary-value problem of ordinary differential equations
\begin{equation}\label{e3.6}
\begin{gathered}
-\xi u_\xi+(u^{2}+\varepsilon \alpha u)_\xi=0, \\
-\xi v_\xi+(uv+\varepsilon \beta u+\varepsilon \gamma v)_\xi=0,
\end{gathered}
\end{equation}
with the boundary condition $(u,v)(\pm \infty)=(u_{\pm},v_{\pm})$.

 Let us denote $U=(u,v)^{T}$ and consider the smooth solutions of the above
boundary-value problem, then \eqref{e3.6} may be rewritten as
 \begin{equation}\label{e3.7}
 A(U)U_{\xi}=0,
 \end{equation}
in which
\begin{equation}\label{e3.8}
A(u,v)=  \begin{pmatrix} {-\xi+2u+\varepsilon\alpha} & {0} \\
 {v+\varepsilon\beta}
 & {-\xi+u+\varepsilon\gamma}
 \end{pmatrix}.
\end{equation}

Besides the constant state solution, it provides a rarefaction wave
which is a continuous solution of \eqref{e3.7} in the form
$(u,v)(\xi)$ which is a function of the single variable
$\xi=\frac{x}{t}$. For a fixed left
state $(u_{-},v_{-})$, the rarefaction curves
in the $(u,v)$ phase plane, which are the sets of states that may be
connected on the right by a rarefaction wave, are as follows:
\begin{equation}\label{e3.9}
R(u_{-},v_{-}):\begin{cases}
\xi=\lambda_2(u,v)=2u+\varepsilon\alpha, \\
\frac{v+\varepsilon\beta}{u+\varepsilon\alpha-\varepsilon\gamma}
=\frac{v_{-}+\varepsilon\beta}{u_{-}+\varepsilon\alpha-\varepsilon\gamma}, \\
u_{-}<u.
\end{cases}
\end{equation}
It is worthwhile to notice that it is a 1-rarefaction wave for
$u_{-}<u<\varepsilon(\gamma-\alpha)$ and  a 2-rarefaction wave for
$\varepsilon(\gamma-\alpha)<u_{-}<u$.

Let us turn to the study of shock wave curves. For a bounded
discontinuity at $x=x(t)$, the Rankine-Hugoniot conditions can be
expressed as
\begin{equation}\label{e3.10}
\begin{gathered}
\sigma [u]=[u^{2}+\varepsilon \alpha u], \\
\sigma [v]=[uv+\varepsilon \beta u+\varepsilon \gamma v],
\end{gathered}
\end{equation}
where $\sigma=\frac{dx}{dt}$ and $[u]=u_{r}-u_{l}$ with
$u_{l}=u(x(t)-0,t)$ and $u_{r}=u(x(t)+0,t)$, etc.

It follows from the first equation in \eqref{e3.10} that
\begin{equation}\label{e3.11}
(\sigma-u_{l}-u_{r}-\varepsilon\alpha)(u_{r}-u_{l})=0.
\end{equation}
If  $u_{l}=u_{r}$, then it follows from the second equation in
\eqref{e3.10} that
\begin{equation}\label{e3.12}
\sigma=u_{l}+\varepsilon\gamma=u_{r}+\varepsilon\gamma,
\end{equation}
which responds to the contact discontinuity.  Thus, for a fixed left
state $(u_{-},v_{-})$, the contact discontinuity curves in the
$(u,v)$ phase plane are as follows:
\begin{equation}\label{e3.13}
J(u_{-},v_{-}):\begin{cases}
\tau=u+\varepsilon\gamma=u_{-}+\varepsilon\gamma, \\
u=u_{-}.
\end{cases}
\end{equation}

On the other hand, if $\sigma=u_{l}+u_{r}+\varepsilon\alpha$, then
it follows from the second equation in \eqref{e3.10} that the relation
\begin{equation}\label{e3.14}
\frac{v_{r}-v_{l}}{u_{r}-u_{l}}=\frac{v_{r}+\varepsilon\beta}{u_{r}
+\varepsilon(\alpha-\gamma)}
=\frac{v_{l}+\varepsilon\beta}{u_{l}+\varepsilon(\alpha-\gamma)}
\end{equation}
should be satisfied. It is well known that an admissibility
condition should be added
   in order to rule out non-physical shock waves. Here the Lax entropy conditions deduce that $u<u_{-}<\varepsilon(\gamma-\alpha)$ for the 1-shock wave
   and $\varepsilon(\gamma-\alpha)<u<u_{-}$ for the 2-shock wave
   should be satisfied. Through the above analysis, for a given left state
 $(u_{-},v_{-})$, the shock
curves in the $(u,v)$ phase plane are as follows:
\begin{equation}\label{e3.15}
S(u_{-},v_{-}):\begin{cases}
\sigma=u_{-}+u+\varepsilon\alpha, \\
\frac{v+\varepsilon\beta}{u+\varepsilon \alpha-\varepsilon\gamma}
=\frac{v_{-}+\varepsilon\beta}{u_{-}+\varepsilon\alpha-\varepsilon\gamma}, \\
u<u_{-}.
\end{cases}
\end{equation}
It is worthwhile to notice that the shock curves coincide with the
rarefaction curves in the $(u,v)$ phase plane, thus  \eqref{e1.2}
belongs to the so-called Temple class \cite{B.Temple}.

Using the elementary waves discussed above, one is in a position to
construct the solutions of the Riemann problem \eqref{e1.2} and \eqref{e1.3} in
the following six different cases according to the values of $u_{-}$
and $u_{+}$.

(1)  If $\varepsilon(\gamma-\alpha)<u_{+}<u_{-}$, then the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} is $J+S$ and the intermediate state
between $J$ and $S$ is determined by
\begin{equation}\label{e3.16}
\begin{gathered}
u_{*}=u_{-}, \\
\frac{v_{*}+\varepsilon\beta}{u_{*}+\varepsilon\alpha-\varepsilon\gamma}
=\frac{v_{+}+\varepsilon\beta}{u_{+}+\varepsilon\alpha-\varepsilon\gamma},
\end{gathered}
\end{equation}
which enables us to have
\begin{equation}\label{e3.17}
(u_{*},v_{*})=\Big(u_{-},v_{+}-\frac{(u_{+}-u_{-})(v_{+}
+\varepsilon\beta)}{u_{+}+\varepsilon
\alpha-\varepsilon \gamma}\Big).
\end{equation}
Thus, when $\varepsilon(\gamma-\alpha)<u_{+}<u_{-}$, the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} can be expressed as:
\begin{equation}\label{e3.18}
(u,v)(x,t)= \begin{cases}
   (u_{-}, v_{-}), &  \xi<\tau_1, \\
(u_{*}, v_{*}), &  \tau_1<\xi<\sigma_2, \\
   (u_{+}, v_{+}),     &  \xi>\sigma_2,
 \end{cases}
\end{equation}
in which the intermediate state $(u_{*}, v_{*})$ is given by \eqref{e3.17}
and the propagation speeds of $J_1$ and $S_2$ can be calculated
 by $\tau_1=u_{-}+\varepsilon\gamma$ and
 $\sigma_2=u_{-}+u_{+}+\varepsilon\alpha$ respectively.

\begin{figure}[htb]
\begin{center}
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\end{picture}
\end{center}
\caption{The Riemann solution of \eqref{e1.2} and \eqref{e1.3} is $J+S$
when $\varepsilon(\gamma-\alpha)<u_{+}<u_{-}$, where $\beta>0$,
$\gamma>\alpha$ and $\varepsilon$ is a sufficiently small positive
number.}
\label{fig1}
\end{figure}

\begin{figure}[htb]
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\put(155.25,32.48){\makebox(0,0)[cc]{\footnotesize$R$}}
\end{picture}
\end{center}
\caption{The Riemann solution of \eqref{e1.2} and \eqref{e1.3} is $J+R$
when $\varepsilon(\gamma-\alpha)<u_{-}<u_{+}$, where $\beta<0$,
$\gamma>\alpha$ and $\varepsilon$ is a sufficiently small positive
number.}
\label{fig2}
\end{figure}

(2)  If $\varepsilon(\gamma-\alpha)<u_{-}<u_{+}$, then the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} is $J+R$ and the intermediate state
between $J$ and $R$ can also be calculated by \eqref{e3.17}. The state
$(u,v)$ in $R_2$ is determined by
\begin{equation}\label{e3.19}
\begin{gathered}
\xi=2u+\varepsilon\alpha, \\
\frac{v+\varepsilon\beta}{u+\varepsilon\alpha-\varepsilon\gamma}
=\frac{v_{+}+\varepsilon\beta}{u_{+}+\varepsilon\alpha-\varepsilon\gamma},
\end{gathered}
\end{equation}
such that we have
\begin{equation}\label{e3.20}
(u,v)=\Big(\frac{\xi-\varepsilon\alpha}{2},v_{+}-\frac{(u_{+}
-\frac{\xi-\varepsilon\alpha}{2})(v_{+}+\varepsilon\beta)}{u_{+}+\varepsilon
\alpha-\varepsilon \gamma}\Big).
\end{equation}
Thus, when $\varepsilon(\gamma-\alpha)<u_{-}<u_{+}$, the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} is
\begin{equation}\label{e3.21}
(u,v)(x,t)= \begin{cases}
   (u_{-}, v_{-}), &  \xi<u_{-}+\varepsilon\gamma, \\
(u_{*}, v_{*}), &  u_{-}+\varepsilon\gamma<\xi<2u_{-}+\varepsilon\alpha, \\
R_2, & 2u_{-}+\varepsilon\alpha \leq \xi \leq
2u_{+}+\varepsilon\alpha, \\
   (u_{+}, v_{+}),     &  \xi>2u_{+}+\varepsilon\alpha,
 \end{cases}
\end{equation}
in which the intermediate state $(u_{*}, v_{*})$ and the state
$(u,v)$ in $R_2$ are given by \eqref{e3.17} and \eqref{e3.20} respectively.

\begin{figure}[htb]
\begin{center}
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\begin{picture}(165.92,50.87)(10,0)
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\end{picture}
\end{center}
\caption{The Riemann solution of \eqref{e1.2} and \eqref{e1.3} is $S+J$
when $u_{+}<u_{-}<\varepsilon(\gamma-\alpha)$, where $\beta>0$,
$\gamma<\alpha$ and $\varepsilon$ is a sufficiently small positive
number.}
\end{figure}

\begin{figure}[htb]
\begin{center}
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\caption{The Riemann solution of \eqref{e1.2} and \eqref{e1.3} is $R+J$
when $u_{-}<u_{+}<\varepsilon(\gamma-\alpha)$, where $\beta<0$,
$\gamma<\alpha$ and $\varepsilon$ is a sufficiently small positive
number.}
\end{figure}


(3)  If $u_{+}<u_{-}<\varepsilon(\gamma-\alpha)$, then the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} is $S+J$ and the intermediate state
between $S$ and $J$ is determined by
\begin{equation}\label{e3.22}
\begin{gathered}
\frac{v_{*}+\varepsilon\beta}{u_{*}+\varepsilon\alpha-\varepsilon\gamma}
=\frac{v_{-}+\varepsilon\beta}{u_{-}+\varepsilon\alpha-\varepsilon\gamma},\\
u_{*}=u_{+},
\end{gathered}
\end{equation}
which implies that
\begin{equation}\label{e3.23}
(u_{*},v_{*})=\Big(u_{+},v_{-}+\frac{(u_{+}-u_{-})(v_{-}
+\varepsilon\beta)}{u_{-}+\varepsilon
\alpha-\varepsilon \gamma}\Big).
\end{equation}
Thus, when $u_{+}<u_{-}<\varepsilon(\gamma-\alpha)$, the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} ie represented as
\begin{equation}\label{e3.24}
(u,v)(x,t)= \begin{cases}
   (u_{-}, v_{-}), &  \xi<\sigma_1, \\
(u_{*}, v_{*}), &  \sigma_1<\xi<\tau_2, \\
   (u_{+}, v_{+}),     &  \xi>\tau_2,
 \end{cases}
\end{equation}
in which the intermediate state $(u_{*}, v_{*})$ is given by \eqref{e3.23}
and the propagation speeds of $S_1$ and $J_2$ can be calculated
 by $\sigma_1=u_{-}+u_{+}+\varepsilon\alpha$ and
$\tau_2=u_{+}+\varepsilon\gamma$ respectively.


 (4)  If $u_{-}<u_{+}<\varepsilon(\gamma-\alpha)$, then the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} is $R+J$ and the intermediate state
between $R$ and $J$ can also be calculated by \eqref{e3.23}. The state
$(u,v)$ in $R_1$ is determined by
\begin{equation}\label{e3.25}
\begin{gathered}
\xi=2u+\varepsilon\alpha, \\
\frac{v+\varepsilon\beta}{u+\varepsilon\alpha-\varepsilon\gamma}
=\frac{v_{-}+\varepsilon\beta}{u_{-}+\varepsilon\alpha-\varepsilon\gamma},
\end{gathered}
\end{equation}
such that we have
\begin{equation}\label{e3.26}
(u,v)=\Big(\frac{\xi-\varepsilon\alpha}{2},v_{-}
-\frac{(u_{-}-\frac{\xi-\varepsilon\alpha}{2})(v_{-}
+\varepsilon\beta)}{u_{-}+\varepsilon
\alpha-\varepsilon \gamma}\Big).
\end{equation}
Thus, when $u_{-}<u_{+}<\varepsilon(\gamma-\alpha)$, the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} is
\begin{equation}\label{e3.27}
(u,v)(x,t)= \begin{cases}
   (u_{-}, v_{-}), &  \xi<2u_{-}+\varepsilon\alpha, \\
R_1, & 2u_{-}+\varepsilon\alpha \leq \xi \leq
2u_{+}+\varepsilon\alpha, \\
(u_{*}, v_{*}), &  2u_{+}+\varepsilon\alpha<\xi<u_{+}+\varepsilon\gamma, \\
   (u_{+}, v_{+}),     &  \xi>u_{+}+\varepsilon\gamma,
 \end{cases}
\end{equation}
in which the state $(u,v)$ in $R_1$ and the intermediate state
$(u_{*}, v_{*})$  are given by \eqref{e3.26} and \eqref{e3.23} respectively.

(5)  If $u_{-}<\varepsilon(\gamma-\alpha)<u_{+}$, then the Riemann
solution to \eqref{e1.2} and \eqref{e1.3} is $R_1+J+R_2$ which can be
expressed as
\begin{equation}\label{e3.28}
(u,v)(x,t)= \begin{cases}
   (u_{-}, v_{-}), &  \xi<2u_{-}+\varepsilon\alpha, \\
R_1, & 2u_{-}+\varepsilon\alpha \leq \xi <
\varepsilon(2\gamma-\alpha), \\
J, & \xi=\varepsilon(2\gamma-\alpha), \\
R_2, &  \varepsilon(2\gamma-\alpha)<\xi\leq 2u_{+}+\varepsilon\alpha, \\
   (u_{+}, v_{+}),     &  \xi>2u_{+}+\varepsilon\alpha,
 \end{cases}
\end{equation}
where the states $(u_1,v_1)$ in $R_1$ and $(u_2,v_2)$ in
$R_2$ are given by \eqref{e3.26} and \eqref{e3.20} respectively. It is
remarkable that the two rarefaction waves $R_1$ and $R_2$ are
connected by the contact discontinuity $J$ directly.

(6)  If $u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$, then one can see
that the singularity is impossible to be a jump with finite
amplitude, which implies that there is no solution which is
piecewise smooth and bounded. Motivated by \cite{D.Tan}, when
$u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$, a solution containing a
weighted delta measure supported on a curve should be introduced
into the solution to the Riemann problem \eqref{e1.2} and \eqref{e1.3}.

For the case $u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$,  it can be concluded
from Definitions \ref{def2.1} and \ref{def2.2}  that the delta shock wave solution
  to the Riemann problem \eqref{e1.2} and \eqref{e1.3}  can also be constructed in
  the form
\begin{equation}  \label{e3.29}
(u,v)(x,t)=\begin{cases}
(u_{-},v_{-}), & \xi<\sigma_{\delta}, \\
(u_{\delta},w(t)\delta (x-\sigma_{\delta} t)), & \xi=\sigma_{\delta}, \\
(u_{+},v_{+}), & \xi>\sigma_{\delta},
\end{cases}
\end{equation}
where $w(t)$ and $\sigma_{\delta}$ denote the strength and
propagation speed of delta shock wave and $u_{\delta}$ indicates the
assignment of $u$ on this delta shock wave curve, respectively. In
fact, the delta shock wave solution in the form \eqref{e3.29}
 to the Riemann problem \eqref{e1.2} and \eqref{e1.3} should also satisfy

\begin{figure}[htb]
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\bezier{300}(47.3,28.06)(53.66,19.29)(60.02,10.51)
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{\footnotesize$(\varepsilon(\gamma-\alpha),-\varepsilon\beta)$}}
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\put(67.02,14.51){\makebox(0,0)[ct]{\footnotesize$(u_+,v_+)$}}
\put(52.14,25.86){\circle*{.6}}
\put(52.14,24.86){\makebox(0,0)[ct]{\footnotesize$0$}}
\put(36.15,35.51){\vector(3,-2){.07}}
\put(54.2,18.56){\vector(3,-4){.07}}
\put(36.95,38.63){\makebox(0,0)[cc]{\footnotesize$R_1$}}
\put(51,18.47){\makebox(0,0)[cc]{\footnotesize$R_2$}}
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\bezier{500}(128.13,6.2)(132.07,24.59)(136.01,42.99)
\bezier{500}(128.13,6.2)(134.54,23.7)(140.95,41.2)
\bezier{500}(128.13,6.2)(137.01,21.81)(145.89,37.42)
\bezier{500}(128.13,6.2)(126.5,24.75)(124.87,43.31)
\bezier{500}(128.13,6.2)(123.72,23.54)(119.3,40.89)
\bezier{500}(128.13,6.2)(120.88,21.75)(113.63,37.31)
\bezier{500}(128.13,6.2)(118.35,19.34)(108.58,32.48)
\bezier{500}(128.13,6.2)(139.06,19.55)(149.99,32.9)
\put(128.13,6.1){\circle*{.6}}
%-
\put(128.13,5.1){\makebox(0,0)[ct]{\footnotesize$0$}}
\linethickness{0.8pt}%
\put(105.95,15.14){\makebox(0,0)[cc]{\footnotesize$(u_-,v_-)$}}
\textcolor[rgb]{0.00,0.00,1.00}{\bezier{500}(128.13,6.2)(129.50,25.75)(130.86,45.3)}
\put(147.47,14.61){\makebox(0,0)[cc]{\footnotesize$(u_+,v_+)$}}
\put(114.05,41.83){\makebox(0,0)[cc]{\footnotesize$R_1$}}
\put(143.9,41.83){\makebox(0,0)[cc]{\footnotesize$R_2$}}
\textcolor[rgb]{0.00,0.00,1.00}{\put(132.65,45.93){\makebox(0,0)[cc]{\footnotesize$J$}}}
\end{picture}
\end{center}
\caption{The Riemann solution of \eqref{e1.2} and \eqref{e1.3} is
$R_1+J+R_2$ when $u_{-}<\varepsilon(\gamma-\alpha)<u_{+}$, where
$\beta<0$, $\gamma<\alpha$ and $\varepsilon$ is a sufficiently small
positive number.}
\label{fig5}
\end{figure}

\begin{figure}[htb]
\begin{center}
\unitlength 0.7mm 
\linethickness{0.4pt}
\ifx\plotpoint\undefined\newsavebox{\plotpoint}\fi % GNUPLOT compatibility
\begin{picture}(165.92,50.72)(10,0)
\put(8.5,43.72){\vector(0,1){.07}} \put(8.5,35.72){\line(0,1){8}}
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\put(95.04,6.12){\line(1,0){69.25}}
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\put(7.87,6.29){\line(1,0){69.25}}
\put(78.76,6.42){\makebox(0,0)[cc]{\footnotesize$u$}}
\put(95.97,42.78){\makebox(0,0)[cc]{\footnotesize$t$}}
\put(165.92,6.22){\makebox(0,0)[cc]{\footnotesize$x$}}
\put(10.22,43.61){\makebox(0,0)[cc]{\footnotesize$v$}}
\put(19.76,39.73){\circle*{.6}}
%-
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{\footnotesize$(\varepsilon(\gamma-\alpha),-\varepsilon\beta)$}}
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%-
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\textcolor[rgb]{1.00,0.00,1.00}{\put(148.58,36.68){\makebox(0,0)[cc]{\footnotesize$\delta
S$}}}%
\put(149.99,14.4){\makebox(0,0)[cc]{\footnotesize$(u_+,v_+)$}}
\end{picture}
\end{center}
\caption{The Riemann solution of \eqref{e1.2} and \eqref{e1.3} is a delta
shock wave $\delta S$ when $u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$,
where $\beta<0$, $\gamma>\alpha$ and $\varepsilon$ is a sufficiently
small positive number.}
\label{fig6}
\end{figure}


\begin{equation}\label{e3.30}
\begin{gathered}
\langle u, \psi_t \rangle+\langle u^{2}+\varepsilon \alpha u, \psi_{x} \rangle=0, \\
\langle v, \psi_t \rangle+\langle u v+\varepsilon \beta
u+\varepsilon \gamma v, \psi_{x} \rangle=0,
\end{gathered}
\end{equation}
for all test functions $\psi(x,t) \in C^{\infty}_0(R \times
R_{+})$. Actually, we have the following theorem to describe
completely the delta shock wave solution
to the Riemann problem \eqref{e1.2} and \eqref{e1.3} for
the case $u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$.

\begin{theorem} \label{thm3.1}
If $u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$, then the delta shock
wave solution to the Riemann problem \eqref{e1.2} and \eqref{e1.3} can be
expressed in the form of \eqref{e3.29} where
\begin{equation}  \label{e3.31}
\begin{gathered}
u_{\delta}=u_{-}+u_{+}+\varepsilon(\alpha-\gamma),  \\
\sigma_{\delta}=u_{-}+u_{+}+\varepsilon \alpha, \\
 w(t)=(u_{-}v_{+}-u_{+}v_{-}+\varepsilon(\alpha-\gamma)(v_{+}-v_{-})
-\varepsilon\beta(u_{+}-u_{-}))t.
\end{gathered}
\end{equation}
Furthermore, the delta shock wave solution \eqref{e3.29} with \eqref{e3.31}
should also satisfy the following generalized Rankine-Hugoniot condition
\begin{equation}  \label{e3.32}
\begin{gathered}
\frac{dx}{dt}=\sigma_{\delta},  \\
\frac{dw(t)}{dt}=\sigma_{\delta}[v]-[uv+\varepsilon \beta u+\varepsilon \gamma v], \\
[u^{2}+\varepsilon \alpha u]=\sigma_{\delta}[u],
\end{gathered}
\end{equation}
and the over-compressive entropy condition
\begin{equation}\label{e3.33}
\lambda_2(u_{+},v_{+})<\lambda_1(u_{+},v_{+})<\sigma_{\delta}
<\lambda_1(u_{-},v_{-})<\lambda_2(u_{-},v_{-}).
\end{equation}
\end{theorem}

\begin{proof}
It is to check that the delta shock wave solution \eqref{e3.29} with \eqref{e3.31}
should satisfy  \eqref{e1.2} in the sense of distributions. Based
on the definition of Schwarz distributions, it is equivalent to
proving that \eqref{e3.29} with \eqref{e3.31} should satisfy
\begin{equation}\label{e3.34}
\begin{gathered}
 \int^{\infty}_0
 \int^{\infty}_{-\infty}\Big(u\psi_t+(u^{2}+\varepsilon \alpha u)\psi_{x}\Big)\,dx\,dt
=0, \\
 \int^{\infty}_0
 \int^{\infty}_{-\infty}\Big(v\psi_t+(uv+\varepsilon \beta u+\varepsilon \gamma
 v)\psi_{x}\Big)\,dx\,dt=0,
\end{gathered}
\end{equation}
which is a weak form of  system \eqref{e1.2}.

Without loss of generality, let us assume $\sigma_{\delta}>0$,
noticing the fact that $\psi$ is compact support in $R^{2}_{+}$,
then we have
\begin{align*}
I&=\int^{\infty}_0
 \int^{\infty}_{-\infty}\Big(u\psi_t+(u^{2}+\varepsilon \alpha u)\psi_{x}\Big)
\,dx\,dt \\
&=\int^{\infty}_0
 \int^{x(t)}_{-\infty}\Big(u_{-}\psi_t+(u_{-}^{2}+\varepsilon \alpha u_{-})
\psi_{x}\Big)\,dx\,dt+\int^{\infty}_0
 \int^{\infty}_{x(t)}\Big(u_{+}\psi_t+(u_{+}^{2}
+\varepsilon \alpha u_{+})\psi_{x}\Big)\,dx\,dt \\
&=\int^{\infty}_0
 \int^{\infty}_{t(x)} u_{-}\psi_tdtdx+\int^{\infty}_0
 \int^{t(x)}_0 u_{+}\psi_tdtdx+\int^{\infty}_0(u_{-}^{2}
 +\varepsilon \alpha u_{-}-u_{+}^{2}-\varepsilon \alpha u_{+})
\psi(x(t),t)dt \\
&=\int^{\infty}_0
(u_{+}-u_{-})\psi(x,t(x))dx+\int^{\infty}_0(u_{-}^{2}+\varepsilon
\alpha u_{-}-u_{+}^{2}-\varepsilon \alpha u_{+}) \psi(x(t),t)dt,
\end{align*}
where $t=t(x)=\frac{x}{\sigma_{\delta}}$ is the inverse function of
$x=x(t)=\sigma_{\delta} t$. Thus, one can arrive at
$\sigma_{\delta}=\frac{dx}{dt}=u_{-}+u_{+}+\varepsilon \alpha$ which
 satisfies $\sigma_{\delta}[u] =[u^{2}+\varepsilon \alpha u]$ for $I$
vanishes for any test function $\psi(x,t)\in C_{c}^{\infty}(R\times
R_{+})$.

Analogously, we also have
\begin{align*}
II&=\int^{\infty}_0
 \int^{\infty}_{-\infty}\Big(v\psi_t+(uv+\varepsilon \beta u+\varepsilon
\gamma v)\psi_{x}\Big)\,dx\,dt \\
&=\int^{\infty}_0
 \int^{x(t)}_{-\infty}\Big(v_{-}\psi_t+(u_{-}v_{-}+\varepsilon \beta u_{-}
 +\varepsilon \gamma  v_{-})\psi_{x}\Big)\,dx\,dt \\
&\quad+\int^{\infty}_0
 \int^{\infty}_{x(t)}\Big(v_{+}\psi_t+(u_{+}v_{+}+\varepsilon \beta u_{+}
 +\varepsilon \gamma  v_{+})\psi_{x}\Big)\,dx\,dt\\
&\quad  +\int^{\infty}_0w(t)\Big( \psi_t(x(t),t)+(u_{\delta}
 +\varepsilon\gamma) \psi_{x}(x(t),t)\Big)dt\\
&= \int^{\infty}_0\Big(u_{-}v_{-}+\varepsilon \beta u_{-}+\varepsilon \gamma
 v_{-}-u_{+}v_{+}-\varepsilon \beta
u_{+}-\varepsilon \gamma
 v_{+}\Big) \psi(x(t),t)dt \\
&\quad  +\int^{\infty}_0
(v_{+}-v_{-})\psi(x,t(x))dx+\int^{\infty}_0w(t)d\psi(x(t),t),
\end{align*}
in which $u_{\delta}+\varepsilon\gamma=\sigma_{\delta}$ has been
used. Thus, one can see that the second equality in \eqref{e3.32} should be
satisfied since $II$ vanishes for any test function $\psi(x,t)\in
C_{c}^{\infty}(R\times R_{+})$, such that one can get the strength
$w(t)$ of delta shock wave.

To ensure the uniqueness of delta shock wave solution to
the Riemann problem \eqref{e1.2} and \eqref{e1.3} when
$u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$, the over-compressive
entropy condition should be proposed. Remember that
$\lambda_1<\lambda_2$ when $u>\varepsilon(\gamma-\alpha)$ and
$\lambda_1>\lambda_2$
 when
$u<\varepsilon(\gamma-\alpha)$. If
$u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$, then the over-compressive
entropy condition for delta shock wave \eqref{e3.33} should be proposed,
which implies that all the characteristics enter
 the delta shock wave curve from both sides.
\end{proof}

\section{Limits of Riemann solutions as $\varepsilon\to0$}

In this section, we are concerned that  the limits  of solutions
to the Riemann problem \eqref{e1.2} and \eqref{e1.3} converge to the corresponding
ones of  the Riemann problem \eqref{e1.1} and \eqref{e1.3} or not when the
perturbation parameter $\varepsilon$ tends to zero. In what follows,
we have the theorem to depict the limit problem fully.

 \begin{theorem} \label{thm4.1}
The limits of solutions to the Riemann problem \eqref{e1.2} with \eqref{e1.3}
converge to the corresponding ones for the non-strictly hyperbolic
system \eqref{e1.1} with the same Riemann initial data as $\varepsilon
\to 0$ in all kinds of situations.
\end{theorem}

\begin{proof}
The proof should also be divided into the following six cases
according to the values of $u_{-}$ and $u_{+}$.

(1) If  $\varepsilon(\gamma-\alpha)<u_{+}<u_{-}$, then it tends to
$0<u_{+}<u_{-}$ by taking the limit $\varepsilon\to0$. It is
clear to see from \eqref{e3.17} and \eqref{e3.18} that the limit
$\varepsilon\to0$ of Riemann solution to \eqref{e1.2} and \eqref{e1.3} is
also $J+S$ which can be expressed as
\begin{equation}\label{e4.1}
\lim_{ \varepsilon \to 0}(u,v)(x,t)
= \begin{cases}
(u_{-}, v_{-}), &  \xi<u_{-}, \\
(u_{-}, \frac{u_{-}v_{+}}{u_{+}}), &  u_{-}<\xi<u_{-}+u_{+}, \\
(u_{+}, v_{+}),     &  \xi>u_{-}+u_{+}.
 \end{cases}
\end{equation}


(2) If  $\varepsilon(\gamma-\alpha)<u_{-}<u_{+}$, then we have
$0<u_{-}<u_{+}$ in the limit situation. Now it follows from \eqref{e3.21}
together with \eqref{e3.17} and \eqref{e3.20} that the limit
$\varepsilon\to0$ of Riemann solution to \eqref{e1.2} and \eqref{e1.3} is
$J+R$ which can be expressed as
\begin{equation}\label{e4.2}
\lim_{\varepsilon \to 0}(u,v)(x,t)
= \begin{cases}
 (u_{-}, v_{-}), &  \xi<u_{-}, \\
(u_{-}, \frac{u_{-}v_{+}}{u_{+}}), &  u_{-}<\xi<2u_{-}, \\
(\frac{\xi}{2},\frac{\xi v_{+}}{2u_{+}}), & 2u_{-} \leq \xi \leq 2u_{+}, \\
   (u_{+}, v_{+}),     &  \xi>2u_{+}.
 \end{cases}
\end{equation}

(3)  If $u_{+}<u_{-}<\varepsilon(\gamma-\alpha)$, then we have
$u_{+}<u_{-}<0$ in the limit situation. Then, it can be derived from
\eqref{e3.23} and \eqref{e3.24} that the limit $\varepsilon\to0$ of
Riemann solution to \eqref{e1.2} and \eqref{e1.3} is also $S+J$ given by
\begin{equation}\label{e4.3}
\lim_{ \varepsilon \to 0}(u,v)(x,t)
= \begin{cases}
   (u_{-}, v_{-}), &  \xi<u_{-}+u_{+}, \\
(u_{+}, \frac{u_{+}v_{-}}{u_{-}}), &  u_{-}+u_{+}<\xi<u_{+}, \\
   (u_{+}, v_{+}),     &  \xi>u_{+}.
 \end{cases}
\end{equation}

(4)  If $u_{-}<u_{+}<\varepsilon(\gamma-\alpha)$, then it is cleat
that $u_{-}<u_{+}<0$ in the limit situation. Then, it follows from
\eqref{e3.27} together with \eqref{e3.23} and \eqref{e3.26} that the limit
$\varepsilon\to0$ of Riemann solution to \eqref{e1.2} and \eqref{e1.3} is
also $R+J$ given by
\begin{equation}\label{e4.4}
\lim_{ \varepsilon \to 0}(u,v)(x,t)
= \begin{cases}
 (u_{-}, v_{-}), &  \xi<2u_{-}, \\
(\frac{\xi}{2},\frac{\xi v_{-}}{2u_{-}}), & 2u_{-} \leq \xi \leq
2u_{+}, \\
(u_{+}, \frac{u_{+}v_{-}}{u_{-}}), &  2u_{+}<\xi<u_{+}, \\
   (u_{+}, v_{+}),     &  \xi>u_{+}.
 \end{cases}
\end{equation}

(5)  It is clear to get $u_{-}<0<u_{+}$ by taking the limit
$\varepsilon\to0$ in the inequality
$u_{-}<\varepsilon(\gamma-\alpha)<u_{+}$. Then, it follows from
\eqref{e3.28} together with \eqref{e3.20} and \eqref{e3.26} that the limit
$\varepsilon\to0$ of Riemann solution to \eqref{e1.2} and \eqref{e1.3} is
also $R+J+R$ given by
\begin{equation}\label{e4.5}
\lim_{ \varepsilon \to 0}(u,v)(x,t)= \begin{cases}
 (u_{-}, v_{-}), &  \xi<2u_{-}, \\
(\frac{\xi}{2},\frac{\xi v_{-}}{2u_{-}}), & 2u_{-} \leq \xi<0, \\
(0,0), &  \xi=0, \\
(\frac{\xi}{2}, \frac{\xi v_{+}}{2u_{+}}), &  0<\xi\leq 2u_{+}, \\
   (u_{+}, v_{+}),     &  \xi>2u_{+}.
 \end{cases}
\end{equation}

(6)  Finally, we also have $u_{+}<0<u_{-}$ by taking the limit
$\varepsilon\to0$ in the inequality
$u_{+}<\varepsilon(\gamma-\alpha)<u_{-}$. Then, it follows from
\eqref{e3.29} and \eqref{e3.31} that the limit $\varepsilon\to0$ of
Riemann solution to \eqref{e1.2} and \eqref{e1.3} is also a delta shock wave given
by
\begin{equation}\label{e4.6}
\begin{aligned}
&\lim_{ \varepsilon \to 0}(u,v)(x,t)\\
&=\begin{cases}
(u_{-},v_{-}), & \xi<u_{-}+u_{+}, \\
(u_{-}+u_{+},(u_{-}v_{+}-u_{+}v_{-})t\delta (x-(u_{-}+u_{+}) t)),
&\xi=u_{-}+u_{+}, \\
(u_{+},v_{+}),\quad \xi>u_{-}+u_{+}.
\end{cases}
\end{aligned}
\end{equation}
Thus, the conclusion of the theorem can be drawn by gathering the
results for the six different cases together.
\end{proof}

\section{Conclusions and discussions}

It can be seen from the above discussions that the limits  of
solutions to the Riemann problem \eqref{e1.2} and \eqref{e1.3} converge to the
corresponding ones of the Riemann problem \eqref{e1.1} and \eqref{e1.3} as
$\varepsilon\to0$. The reason lies in that the approximated
system \eqref{e1.2} is still non-strictly hyperbolic and the characteristic
field for $\lambda_1$
 is still linearly degenerate and the characteristic field for $\lambda_2$
is still genuinely  nonlinear. Thus, this perturbation  does not change the
structure of Riemann solutions.

In addition, let us turn our attentions on the simplified system of
pressureless gas dynamics as follows:
\begin{equation}\label{e5.1}
\begin{gathered}
u_t+(\frac{u^{2}}{2})_x=0, \\
v_t+(uv)_x=0.
\end{gathered}
\end{equation}
It is easy to check that  \eqref{e5.1} has a double eigenvalue
$\lambda=u$ and only one right eigenvector
$\overrightarrow{r}=(0,1)^{T}$. Then, we can get $\nabla \lambda
\cdot \overrightarrow{r}=0$, which implies that the characteristic
field for $\lambda$ is always linearly degenerate. Thus, the
solutions of Riemann problem \eqref{e5.1} and \eqref{e1.3} can be constructed by
contact discontinuities, vacuum or  delta shock wave connecting two
constant states $(u_{\pm}, v_{\pm})$.

If we also consider the linear approximations of flux functions for
 \eqref{e5.1} in the form
\begin{equation}\label{e5.2}
\begin{gathered}
u_t+(\frac{u^{2}}{2}+\varepsilon \alpha u)_x=0, \\
v_t+(uv+\varepsilon \beta u+\varepsilon \gamma v)_x=0.
\end{gathered}
\end{equation}
We can check that if $\alpha=\gamma$, then  \eqref{e5.2} has also
a double eigenvalue $\lambda=u+\varepsilon \alpha$ and only one right
eigenvector $\overrightarrow{r}=(0,1)^{T}$. Furthermore, one can see
that the characteristic field for $\lambda$ is always linearly
degenerate. In other words,  \eqref{e5.2} also belongs to the Temple
class when $\alpha=\gamma$. Through a simple calculation, it can be
concluded that the Riemann solutions for the approximated system \eqref{e5.2} just
translate the ones for the original system \eqref{e5.1} in the $(x,t)$
plane and do not change the structure. Thus, when $\alpha=\gamma$,
we can also see that the limits  of solutions to the Riemann problem
\eqref{e5.2} and \eqref{e1.3} converge to the corresponding ones of the Riemann
problem \eqref{e5.1} and \eqref{e1.3} as $\varepsilon\to0$ in all the
situations.

Otherwise, if $\alpha\neq\gamma$, then  \eqref{e5.2} has two
different eigenvalues $\lambda_1=u+\varepsilon\alpha$ and
$\lambda_2=u+\varepsilon\gamma$. Thus,  \eqref{e5.2} is strictly
hyperbolic for $\alpha\neq\gamma$. It is easy to get that the right
eigenvectors are
$\overrightarrow{r}_1=(\varepsilon(\alpha-\gamma),v+\varepsilon\beta)^{T}$ and
$\overrightarrow{r}_2=(0,1)^{T}$ respectively, such that we have
$\nabla \lambda_1 \cdot \overrightarrow{r}_1=\varepsilon(\alpha-\gamma) \neq
0$ and $\nabla \lambda_2 \cdot \overrightarrow{r}_2=0$. Hence,
the characteristic field for $\lambda_1$ is  genuinely nonlinear
and the characteristic field for $\lambda_2$ is linearly
degenerate.  By a simple calculation, it can be seen that  \eqref{e5.2}
does not belong to the Temple
class any more when $\alpha\neq\gamma$. It is clear to see that if
$\alpha\neq\gamma$, then the
Riemann solutions for the approximated system \eqref{e5.2} have completely
different structures with those for the original system \eqref{e5.1}.
If we introduce the substitutions of state variables $u+\varepsilon\alpha\to u$
and $v+\varepsilon \beta\to v$,
then  \eqref{e5.2} becomes
\begin{equation}\label{e5.3}
\begin{gathered}
u_t+(\frac{u^{2}}{2})_x=0, \\
v_t+(uv+\varepsilon(\gamma-\alpha)v)_x=0.
\end{gathered}
\end{equation}
It is well known that this form of the system does
not change for the reason the variable substitutions are linear in the conserved
quantities.
If $\gamma<\alpha$, then it can be concluded from \cite{C.Shen3} that the
limits of solutions to the Riemann problem \eqref{e5.3} and \eqref{e1.3} also
converge to the corresponding ones of the Riemann problem \eqref{e5.1} and
\eqref{e1.3} as $\varepsilon\to0$ in all the situations. On the other hand,
if $\gamma>\alpha$, then
the similar calculation can be carried out and the same conclusion can be drawn.
 Thus, if $\alpha\neq\gamma$, then
one can conclude that the limits of solutions to the Riemann problem
\eqref{e5.2} and \eqref{e1.3} converge to the corresponding ones of the Riemann
problem \eqref{e5.1} and \eqref{e1.3} as $\varepsilon\to0$ in all the
situations.

\subsection*{Acknowledgments}
This work is partially supported by the National Natural Science Foundation
of China (11271176).

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\end{thebibliography}

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