\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 $u_{+}0$ 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 0v_{-}.
\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_{-}(2u_{+}+1)t.
\end{cases}
\end{equation}
For the case $u_{-}-1(2u_{+}+1)t,
\end{cases}
\end{equation}
where
\begin{equation}\label{e2.5}
v_{*}=v_{-}\cdot\frac{u_{-}-u_{+}+1}{u_{+}-u_{-}+1}.
\end{equation}
\begin{figure}[ht]
\begin{center}
\unitlength 0.9mm
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\put(47.47,41.52){\makebox(0,0)[cc]{$\scriptstyle v$}}
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\put(124.25,7.38){\makebox(0,0)[cc]{$\scriptstyle u$}}
\textcolor[rgb]{0.00,0.00,1.00}{\bezier{500}(87.5,6)(87.5,24.315)(87.5,42.63)}
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\textcolor[rgb]{0.00,0.00,1.00}{\put(89.75,40.88){\makebox(0,0)[cc]{\footnotesize$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)
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\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~~\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_{-}-1x_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_{-}),& xx_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}
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\put(43.99,-0.31){\makebox(0,0)[cc]{(a) $u_--2t_4.
\end{gather}
Otherwise, if $u_{-}-1u_-$}}
\end{picture}
\end{center}
\caption{Interaction between $R+J$ and $S+J$
for two situations where both $u_{-}u_-$, 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 $u_{-}-1u_-$, 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_{-}-10,
\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_{-}-10,
\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]
\begin{center}
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\end{center}
\caption{Interactions between $S+J$ and $R+J$
for two different situations when both $u_{-}-1u_-$, 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 $u_{-}-1u_-$, 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_{-}x_1,
\end{cases}
\qquad
v|_{t=t_1}
=\beta(t_1) \delta_{(x_1,t_1)}
+\begin{cases}
v_{-}, & xx_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_{+}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_{-}-1x_2,
\end{cases}
\qquad
v|_{t=t_2}=\beta(t_2) \delta_{(x_2,t_2)}+
\begin{cases}
v_{-}, & xx_2.
\end{cases}
\end{equation}
We can obtain $u_{+}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_{-}, & xx_1,
\end{cases}
\qquad
v|_{t=t_1}= \beta(t_1) \delta_{(x_1,t_1)}+
\begin{cases}
v_{-}, & xx_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 $u_{m}0$, 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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\end{center}
\caption{Interactions between $\delta S$ and $R+J$
for two different situations where both $u_{m}\leq u_{-}-1$
and $u_{m}u_{-}$, 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
$u_{-}u_{+}+1$, 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 $u_{-}u_-$}}
\put(119.2,0.31){\makebox(0,0)[cc]{(b) $u_--1u_{-}$, 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 $u_{-}-1u_{-}$.
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}
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