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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 94, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/94\hfil Weak asymptotic solution]
{Weak asymptotic solution for a non-strictly hyperbolic system of
conservation laws-II}

\author[M. R. Sahoo, H. Singh \hfil EJDE-2016/94\hfilneg]
{Manas Ranjan Sahoo, Harendra Singh}

\address{Manas Ranjan Sahoo \newline
School of Mathematical Sciences,
National Institute of Scince Education and Research,
Jatni, Khurda-752050, India}
\email{manas@niser.ac.in}

\address{Harendra Singh \newline
Department of Mathematical Sciences,
IIT (BHU),
Varanasi 221005, India}
\email{harendrasingh.rs.apm12@iitbhu.ac.in}

\thanks{Submitted March 14, 2016. Published April 7, 2016.}
\subjclass[2010]{35A20, 35L50, 35R05}
\keywords{System of PDEs; initial condition; weak asymptotic solution}

\begin{abstract}
 In this article we introduce a concept of entropy weak asymptotic
 solution for a system of conservation laws and construct the same
 for a prolonged system of conservation laws which is highly non-strictly
 hyperbolic. This is first done for Riemann type initial data by
 introducing $\delta,\delta'$ and $\delta''$ waves along a discontinuity
 curve and then for general initial data by piecing together the
 Riemann solutions.
\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}

The general theory of systems of conservation laws assumes systems to be
strictly hyperbolic, see; Lax \cite{la1}, Glimm \cite{G1} and Bressan \cite{b1}.
In general for non-strictly hyperbolic system, the solution does not lie
in the class $L^p$, $1 \leq p <\infty$. For existence and uniqueness
result for a particular system which is not strictly hyperbolic we refer
Floch \cite{le}.
In general one has to admit solution space as the space of distributions. Then
due to the appearance of product of distributions, it is difficult to define
the notion of product. One way to overcome this is to work with the generalized
space of Colmbeau,  for details see \cite{co2,ob1}.

Recently a new notion of solution is introduced by Panov and Shelkovich \cite{p1},
called weak asymptotic solution. We would also like to cite another concept of
solution by Marko Nedelkjov \cite{m2}, called shadow wave solution.
This is same as the \emph{solution in the sense of association}
introduced by Colombeau but author also allows non smooth functions.


We study the following system of conservation laws which is studied
in \cite{j2,m2,s1,s2,hm1}; for different values of $n$,  namely;
\begin{equation}
(u_j)_t+\sum_{i=1}^{j}(\frac{u_i u_{j-i+1}}{2})_x=0,\quad j=1,2,\dots, n.
\label{*}
\end{equation}
For $n=1$, system \eqref{*} is Burger's equation, which is well studied
by Hopf \cite{h1}.
For $n=2$ case, \eqref{*} is a one dimensional model for the large scale
structure formation of universe, see \cite{w1}.
Using vanishing viscosity approach it is observed by Joseph \cite{j1}
that the second component contain $\delta$ measure concentrated along
the line of discontinuity.
The case $n=3$ is studied in \cite{j3} in Colombeau setting.
If $u_1=u$, $u_2=v$, $u_3=w$, \eqref{*} becomes
\begin{equation}
u_t +(\frac{u^2}{2})_x=0,\quad  v_t+(uv)_x=0,\quad  w_t+(\frac{v^2}{2} +uw)_x=0.
\label{*2}
\end{equation}
A system similar to the one above, namely
\begin{equation}
u_t +(u^2)_x=0,\quad  v_t+(2uv)_x=0,\quad w_t+2(v^2 +uw)_x=0.
\label{*3}
\end{equation}
is studied in \cite{s1}. There, a concept of weak asymptotic
solution is introduced and using this a generalized integral formulation
is given. Note that the system \eqref{*3} can be obtained from \eqref{*2}
using the transformation $(u,v,w)\to (2u,v,\frac{w}{2})$. The case $n=4$ is studied
by Joseph and Sahoo \cite{jm1}, using vanishing viscosity approach.
In that paper a solution is constructed for Riemann type initial data
and based on  this a weak integral formulation is given.
In \cite{hm1}, weak asymptotic solution is constructed for the
case $n=4$ when $u$ develops shock and initial data are of Riemann type.
Using this,  weak asymptotic solution is constructed for the case $n=4$
 when the initial data for the first component is monotonic increasing and
initial data for the other components are of general type.

In this paper we define entropy weak asymptotic solution for a general
conservation law and construct the same
for the case $n=4$, for Riemann type initial data. Using Riemann type
solutions we construct entropy weak asymptotic solution for some
special general initial data.

For $n=4$, $u_1=u, u_2=v, u_3=w, u_4=z$ and followed by a linear
transformation, the system \eqref{*} leads to the system
\begin{equation}
\begin{gathered}
u_t +(u^2)_x=0,\quad  v_t+(2uv)_x=0\\
w_t+2(v^2 +uw)_x=0,\quad z_t+ 2((3vw+uz)_x)=0.
\end{gathered} \label{e1.2}
\end{equation}
We take initial conditions as
\begin{equation}
u(x,0)=u_0 (x),\quad v(x,0)=v_0 (x), \quad w(x,0)=w_0 (x), \quad z(x,0)=z_0 (x).
\label{e1.3}
\end{equation}
The contents of this article is as follows.
 We define entropy weak asymptotic solution for any general system
of conservation laws.
For Riemann type data we construct the same for the system \eqref{e1.2}
by using asymptotic analysis of the product of regularized singular waves,
namely $\delta, \delta'$ and $\delta''$.
Then we construct a weak asymptotic solution when the initial data for
$u$ is a monotonic function and initial data for $v,w$ and $z$ are
bounded measurable functions.

\section{Weak asymptotic solution for Riemann type initial data}

First of all we define weak asymptotic solution \cite{s1, s2}
and entropy weak asymptotic solution as follows.

\begin{definition} \label{def2.1}\rm
A sequence of smooth functions $u_j ^ {\epsilon}(x,t)$, $j=1,2,dots n$;
is said to be weak asymptotic solution to the system of conservation laws
\begin{equation}
\begin{gathered}
(u_j)_t + (f_j (u_1, u_2...u_n))_x =0\\
u_j (x,0)=u_{0j}(x)
\end{gathered}
\end{equation}
if the following identity hold
\begin{equation}
\begin{gathered}
 {\lim_{\epsilon \to 0}}\int_{-\infty}^ \infty \big[(u_j^ {\epsilon})_t
+ (f_j (u_1^ {\epsilon}, u^ {\epsilon}_2,\dots,u^ {\epsilon}_n))_x
\big]\psi(x)dx =0\\
 {\lim_{\epsilon \to 0}}\int_{-\infty}^ \infty \big[u_j (x,0)-u_{0j}(x)\big]
\psi(x)dx =0
\end{gathered}
\end{equation}
uniformly in the variable $t$ lying in any compact subset of $(0, \infty)$
and for all test function $\psi \in C_c ^ {\infty}(R)$.
\end{definition}

\begin{definition} \label{def2.2} \rm
A sequence of smooth functions $u_j ^ {\epsilon}(x,t)$, $j=1,2,\dots,n$
is said to be entropy admissible weak asymptotic solution
to the following system of conservation laws
\begin{equation}
\begin{gathered}
(u_j)_t + (f_j (u_1, u_2,\dots,u_n))_x =0\\
u_j (x,0)=u_{0j}(x)
\end{gathered}
\end{equation}
if it is a weak asymptotic solution with the following extra condition:
For any $\eta, q \in C^2(\mathbb{R}^n; \mathbb{R})$, with $\eta$
convex and $D\eta(u)DF(u)= Dq(u)$,
\begin{equation}
  {\limsup_{\epsilon \to 0}}\int_{-\infty}^ \infty
\big[\eta((U^ {\epsilon})_t + (q (U^ {\epsilon}))_x\big]\psi(x)dx\leq 0,
\end{equation}
for all test functions $0\leq \psi \in C_c ^ {\infty}(R)$.
Here $U^{\epsilon}= (u_1^ {\epsilon}, u^ {\epsilon}_2,\dots, u^ {\epsilon}_n)$
and $F(u)=(f_1(u), \dots,f_n(u))$.
\end{definition}

In accordance with the above definition, let us define
 \begin{gather*}
L_1 (u)=u_t +(u^2)_x,\quad  L_2 (u,v)= v_t+(2uv)_x\\
L_3 (u,v,w)=w_t+2(v^2 +uw)_x,\quad L_4 (u,v,w,z)=z_t+ 2((3vw+uz)_x).
%\label{e2.1}
\end{gather*}

The expression $(u^{\epsilon}, v^{\epsilon}, w^{\epsilon}, z^{\epsilon})$
is said to be weak asymptotic solution to  \eqref{e1.2} with
initial data \eqref{e1.3} if
\begin{equation}
\begin{gathered}
\int L_1[u(x,t,\epsilon)]\psi(x)dx=o (1)\quad
\int L_2[u(x,t,\epsilon),v(x,t,\epsilon)]\psi(x)dx=o (1)\\
\int L_3[u(x,t,\epsilon),v(x,t,\epsilon),w(x,t,\epsilon)]\psi(x)dx=o (1)\\
\int L_4[u(x,t,\epsilon),v(x,t,\epsilon),w(x,t,\epsilon),z(x,t,\epsilon)]\psi(x)dx
 =o(1),
\end{gathered}\label{e2.2}
\end{equation}
and initial conditions  satisfy
\begin{gather*}
\int\Big(u(x,0,\epsilon)-u_0 (x)\Big)\psi(x)dx=o(1),\quad
\int\Big(v(x,0,\epsilon)-v_0 (x)\Big)\psi(x)dx=o(1)\\
\int\Big(w(x,0,\epsilon)-w_0 (x)\Big)\psi(x)dx=o(1),\quad
\int\Big(z(x,0,\epsilon)-z_0 (x)\Big)\psi(x)dx=o(1),
\end{gather*} %\label{e2.3}
for all $\psi \in D(R)$.

With the similar lines we can have entropy weak asymptotic solution.
To study weak asymptotic analysis first we need the following
Lemma as in \cite{s1}, regarding the superpositions of singular waves;
$\delta,\delta',\delta''$ and $ \delta'''$.

\begin{lemma} \label{lem2.3}
Let $\{w_i \}_{i\in I}$ is an indexed family of Friedrich mollifiers satisfying
$$
w_i (x)=w_i (-x)\quad\text{and}\quad  \int w_i =1.
$$
Define $ H_i(x,\epsilon)=w_{0i}(\frac{x}{\epsilon})=\int_{-
\infty}^{\frac{x}{\epsilon}}w_i(y)d y$,
 $\delta_i (x,\epsilon)=\frac{1}{\epsilon}w_i(\frac{x}{\epsilon})$ and
$\delta^{\prime \prime...\text{(k times)}}_i(x,\epsilon)
=\frac{1}{\epsilon^{k+1}}w_i ^{\prime \prime...\text{(k times)}}
(\frac{x}{\epsilon})$.
The above assumptions implies the following asymptotic
expansions, in the sense of distribution.
\begin{gather*}
(H_i(x,\epsilon))^{r}=H(x)+O_{D'}(\epsilon),\quad
(H_i(x,\epsilon)(H_j(x,\epsilon))=H(x)+O_{D'}(\epsilon)\\
(H_i(x,\epsilon))^{r}\delta_j(x,\epsilon)=\delta(x)\int{w_{0i}^{r}(y)w_j(y)d
y}+O_{D'}(\epsilon)\\
(\delta_i(x,\epsilon))^{2}=\frac{1}{\epsilon}\delta(x)\int{w_i ^{2}(y)d
y}+O_{D'}(\epsilon)\\
H_i(x,\epsilon)\delta_j '(x,\epsilon)
=-\frac{1}{\epsilon}\delta(x)\int{w_{i}(y)w_j (y)d y}
+\delta'(x)\int{w_{0i}(y)w_j(y)d y}+O_{D'}(\epsilon)\\
H_i(x,\epsilon)\epsilon^{2}\delta_j'''(x,\epsilon)
=\frac{1}{\epsilon}\delta(x)\int{w_i'(y)\delta_j' (y)d y}+O_{D'}(\epsilon),\\
\delta_i (x,\epsilon).\delta_j (x,\epsilon)
 =\frac{1}{\epsilon}\delta(x)\int{w_i(y)w_j(y)dy}+O_{D'}(\epsilon)\\
\delta_i (x,\epsilon)\delta_j'(x,\epsilon)=\frac{1}{\epsilon}\delta'(x)\int
{y w_i(y)w_j'(y)d y}+O_{D'}(\epsilon),\\
H_i(x,\epsilon)\delta_j''(x,\epsilon)
=\frac{1}{\epsilon}\delta(x)\int {w_{0i}(y)w_j(y)d y}
 +\frac{1}{2}\delta''(x)\int {y^{2}w_{0i}(y)w_j(y)d y}+O_{D'}(\epsilon)\\
\delta_i(x,\epsilon))\epsilon^{2}\delta_j'''(x,\epsilon)=\frac{1}{\epsilon}\delta'(x)
\int y w_{i}(y) w_j'''(y) dy+O_{D'}(\epsilon)\\
H_i(x,\epsilon)\epsilon^{2}\delta_j''''(x,\epsilon)
=\frac{1}{\epsilon}\delta'(x)
\int y w_{0i}(y) w_j''''(y) dy+O_{D'}(\epsilon),
\end{gather*} %\label{e2.4}
where $ O_{D'}(\epsilon)$ is the error term satisfying
${\lim_{\epsilon \to 0} \langle O_{D'}(\epsilon), \psi(x)\rangle=0} $,
for any test function $ \psi $.
\end{lemma}

\begin{proof}
Let $\psi \in D(\mathbb{R})$ be any test function.
Relations from $1-6$ can be found in \cite{s1}. We prove the
 asymptotic expansions from the seventh onward.

Now we prove the seventh asymptotic expansion.
Using change of variable formula $(x=\epsilon y)$,
employing third order Taylor expansion,
$\psi(\epsilon y)=\psi(0)+\epsilon y\psi' (0)
+\epsilon^2 y^2 \psi''(0)+\epsilon^3 y^3 O(1)$ and the fact that
$\int y w_{i}(y)w_{j}(y)d y=0$, we have
\begin{align*}
\langle\delta_i(x,\epsilon)\delta_j(x,\epsilon),\psi(x)\rangle
&=\int\frac{1}{\epsilon}w_{i}(\frac{x}{\epsilon})
 \frac{1}{\epsilon}w_{j}(\frac{x}{\epsilon})\psi(x)d x \\
& =\frac{1}{\epsilon}\int w_{i}(y)w_{j}(y)\psi(\epsilon y)d y  \\
& =\frac{1}{\epsilon}\psi(0)\int w_{i}(y)w_{j}(y)d
y+\psi'(0)\int y w_{i}(y)w_{j}(y)d y+O(\epsilon)\\
&=\frac{1}{\epsilon}\delta(x)\int { w_i(y)w_j(y)d y}+O(\epsilon).
\end{align*}

Now we prove the eighth asymptotic expansion. Using change of variable
formula $(x=\epsilon y)$, employing third order Taylor expansion,
$\psi(\epsilon y)=\psi(0)+\epsilon y\psi' (0)+\epsilon^2
y^2 \psi''(0)+\epsilon^3 y^3 O(1)$ and the fact that
$\int y w_{i}(y)w_{j}(y)d y=0$, we have
\begin{align*}
\langle\delta_i(x,\epsilon)\delta_j'(x,\epsilon),\psi(x)\rangle
&=\frac{1}{\epsilon^2}\int w_{i}(y)w_{j}'(y)\psi(\epsilon y)d y \\
&=\frac{1}{\epsilon^2}\psi(0)\int w_{i}(y)w_{j}'(y)d
y+\frac{1}{\epsilon}\psi'(0)\int y w_{i}(y)w_{j}'(y)dy\\
&\quad +\frac{1}{2}\psi''(0)\int y^{2} w_{i}(y)w_{j}'(y)d y +O(\epsilon)\\
&=\frac{1}{\epsilon}\delta'(x)\int {y w_i(y)w_j'(y)d
y}+O(\epsilon).
\end{align*}
In the above calculation we also used the identity
$$
\int w_{i}(y)w_{j}'(y)dy=\int {y^2 w_i(y)w_j'(y)dy}=0.
$$

Following the analysis similar as above, we prove the remaining identities.
Details are as follows:
\begin{align*}
&\langle H_i(x,\epsilon)\delta_j''(x,\epsilon),\psi(x)\rangle\\
&=\int w_{0i}(y)\frac{1}{\epsilon^2}w_{j}''(y)\psi(\epsilon y)d y \\
&=\int w_{0i}(y)\frac{1}{\epsilon^2}w_{j}''(y)(\psi(0)+\epsilon y
\psi'(0)+\frac{\epsilon^{2} y^2}{2} \psi''(0))d y+O(\epsilon) \\
&=\frac{1}{\epsilon}\delta'(x)\int {y w_{0i}(y)w_j''(y)d
y}+\frac{1}{2}\delta''(x)\int {y^{2}w_{0i}(y)w_j''(y)d
y}+O(\epsilon).
\end{align*}
\begin{align*}
&\langle \delta_i(x,\epsilon))\epsilon^{2}
 w_j'''(x,\epsilon),\psi(x)\rangle\\
&=\frac{1}{\epsilon^{2}}\int w_{i}(y) w_j'''(y)\psi(\epsilon y)d y \\
&=\frac{1}{\epsilon^{2}}\int w_{i}(y) w_j'''(y)(\psi(0)+\epsilon y
\psi'(0)+\frac{\epsilon^{2} y^2}{2} \psi''(0))d y+O(\epsilon) \\
&=\frac{1}{\epsilon^{2}}\delta(x)\int w_{i}(y) w_j'''(y)dy
 +{\frac{1}{\epsilon}\delta'(x) \int y w_{i}(y) w_j'''(y) dy }\\
&\quad +\frac{1}{2}\delta''(x)\int  y^2
w_{i}(y) w_j'''(y) dy +O(\epsilon)\\
&={\frac{1}{\epsilon}\delta'(x) \int y w_{i}(y) w_j'''(y) dy
}+O(\epsilon).
\end{align*}
\begin{align*}
&\langle H_i(x,\epsilon)\epsilon^2\delta_j''''(x,\epsilon),\psi(x)\rangle\\
&=\int w_{0i}(y)\frac{1}{\epsilon^2}w_{j}''''(y)\psi(\epsilon y)(\psi(0)+\epsilon y
\psi'(0)+\frac{\epsilon^{2} y^2}{2} \psi''(0))d y +O(\epsilon)\\
&=\frac{1}{\epsilon}\delta'(x)
\int y w_{0i}(y) w_j''''(y) dy+O(\epsilon).
\end{align*}
\end{proof}

In \cite{jm1}, it is observed that the vanishing viscosity limit for
the component $z$ admits combinations of $ \delta$ and $\delta'$
waves when $u$ develops rarefaction. So we choose ansatz as the combination
of the above singular waves along the discontinuity curve.
But this is not enough as it is clear in the construction of $w$, see \cite{s1}.
In \cite{s1}, a correction term is added in the component $w$ to construct
weak asymptotic solution for the shock case. Here in this rarefaction case
we will add correction terms in the component $z$, but no correction terms
in the third component $w$. Note that this correction term is required to balance
the unexpected term coming in the product of singular waves.

\begin{theorem} \label{thm2.4}
The ansatz
\begin{equation}
\begin{gathered}
\begin{aligned}
u(x,t,\epsilon)
&=u_1H_{u}(-x+2u_1t,\epsilon)+\frac{x}{2t}\{H_{u}(-x+2u_2t,\epsilon)\\
&\quad -H_{u}(-x+2u_1t,\epsilon)\}
 +u_2\{1-H_{u}(-x+2u_2t,\epsilon)\}.
\end{aligned}\\
v(x,t,\epsilon)=v_1H_{v}(-x+2u_1t,\epsilon)+v_2\{1-H_{v}(-x+2u_2t,\epsilon)\}.
\\
\begin{aligned}
w(x,t,\epsilon)
&=w_1H_{w}(-x+2u_1t,\epsilon)+w_2\{1-H_{w}(-x+2u_2t,\epsilon)\}\\
&\quad +e_1(t)\delta_e(-x+2u_1t,\epsilon)+e_2(t)\delta_e(-x+2u_2t,\epsilon).
\end{aligned} \\
\begin{aligned}
z(x,t,\epsilon)&=z_1H_{z}(-x+2u_1t,\epsilon)+z_2\{1-H_{z}(-x+2u_2t,\epsilon)\}\\
&\quad +g_1(t)\delta_e(-x+2u_1t,\epsilon)
  +g_2(t)\delta_e(-x+2u_2t,\epsilon)
\\
&\quad +h_1(t)\delta'_{g}(-x+2u_1t,\epsilon)+h_2(t)\delta'_{g}(-x+2u_2t,\epsilon)\\
&\quad +R_{1} (-x+2u_1t,\epsilon)+R_{2} (-x+2u_2t,\epsilon),
\end{aligned}
\end{gathered} \label{e2.5}
\end{equation}
where
\begin{gather*}
R_{1} (-x+2u_1t,\epsilon)=\epsilon^2 P_1 (t) \delta_P'''(-x+2u_1t,\epsilon),\\
R_{2} (-x+2u_2t,\epsilon)=\epsilon^2 P_2(t) \delta_P'''(-x+2u_2t,\epsilon).
\end{gather*}
is entropy weak asymptotic solution to the problem \eqref{e1.2} if
the following relations hold.
\begin{equation}
\begin{gathered}
\dot{e_1}(t)=2v_1^2, \quad  \dot{e_2}(t)=-2v_2^2,\\
\int {w_{0u}(y)w_e(y)dy}=\int {w_{0v}(y)w_e(y)dy}=\int {w_{0u}(y)w_g(y)dy}=1,\\
\dot{g_1}(t)=6v_1w_1,\quad \dot{g_2}(t)=-6v_2w_2,\\
\dot{h_1}(t)=6v_1e_1(t), \quad \dot{h_2}(t)=6v_2e_2(t), \\
\int {w_{u}(y)w_g(y)dy}=1, \quad \int {w'_{u}(y)w'_P(y)dy}=1 \\
 P_1(t)=h_1(t),  \quad  P_2(t)=h_2(t).
\end{gathered} \label{e2.6}
\end{equation}
\end{theorem}


\begin{remark} \label{rmk2.5} \rm
 In Theorem \ref{thm2.4}, $R_1$ and $R_2$ are called correction terms
 which is needed to adjust the odd terms appearing in the asymptotic
expansion of the product of   regularized singular waves.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm2.4}]
Using lemma \ref{lem2.3}, we obtain
\begin{equation}
\begin{aligned}
&u^2(x,t,\epsilon)\\
& \approx u_1^2H(-x+2u_1t,\epsilon)+\frac{x^2}{4t^2}
\{H(-x+2u_2t,\epsilon)-H(-x+2u_1t,\epsilon)\}\\&+u_2^2\{1-H(-x+2u_2t,\epsilon)\}.
\label{n1}
\end{aligned}
\end{equation} 
Taking the distributional derivative of $u$ with respect to $t$, we obtain
\begin{equation}
\begin{aligned}
u_t(x,t,\epsilon)
&\approx \frac{x}{2t^2}\{H(-x+2u_1t,\epsilon)
 - H(-x+2u_2t,\epsilon)\}\\
&\quad + \{2u_1^2-\frac{u_1x}{t}\}\delta(-x+2u_1t,\epsilon)
-\{2u_2^2-\frac{u_2x}{t}\}\delta(-x+2u_2t,\epsilon).
\end{aligned} \label{n2}
\end{equation} 
Putting  \eqref{n1} and \eqref{n2} in the first equation of \eqref{e1.2},
we obtain
\begin{align*}
&u_t(x,t,\epsilon)+(u^2)_x(x,t,\epsilon)\\
& \approx \{u_1^2-\frac{u_1x}{t}+\frac{x^2}{4t^2}\}\delta(-x+2u_1t,\epsilon)
-\{u_2^2-\frac{u_2x}{t}+\frac{x^2}{4t^2}\}
\delta(-x+2u_2t,\epsilon)=0.
\end{align*}
Now we find the asymptotic expansion of terms appearing in the second
 equation of \eqref{e1.2}. Applying lemma \ref{lem2.3},
\begin{equation}
\begin{gathered}
u(x,t,\epsilon)v(x,t,\epsilon)
 \approx u_1v_1H(-x+2u_1t,\epsilon)+u_2v_2\{1-H(-x+2u_2t,\epsilon)\}.
\\
v_t(x,t,\epsilon) \approx 2u_1v_1\delta(-x+2u_1t,\epsilon)
-2u_2v_2\delta(-x+2u_2t,\epsilon).
\end{gathered} \label{n3}
\end{equation}
Using  expansions \eqref{n3} in the second equation of \eqref{e1.2}, we obtain
\begin{align*}
&v_t(x,t,\epsilon)+(2u(x,t,\epsilon) v(x,t,\epsilon))_x\\
&\approx \{2u_1v_1-2u_1v_1\}\delta(-x+2u_1t,\epsilon)
 +\{2u_2v_2-2u_2v_2\}\delta(-x+2u_2t,\epsilon)=0
\end{align*} 
Now we find the asymptotic expansion of terms appearing in the third equation
of \eqref{e1.2}. Applying lemma \ref{lem2.3},
\begin{gather}
v^2(x,t,\epsilon)
 \approx v_1^2 H(-x+2u_1t,\epsilon)+v_2^2\{1-H(-x+2u_2t,\epsilon)\}.\label{n4}
\\
\begin{aligned}
&u(x,t,\epsilon)w(x,t,\epsilon)\\
& \approx u_1w_1H(-x+2u_1t,\epsilon)+u_2w_2\{1-H(-x+2u_2t,\epsilon)\}\\
&\quad +u_1e_1(t)\int{w_{0u}(y)w_e(y)dy}\delta(-x+2u_1t,\epsilon)\\
&\quad +u_2e_2(t)\int {w_{0u}(y)w_e(y)dy}\delta(-x+2u_2t,\epsilon),
\end{aligned} \label{n5}
\end{gather} 
Using \eqref{n4} and \eqref{n5}, we obtain
\begin{equation}
\begin{aligned}
&2(v^2(x,t,\epsilon)+u(x,t,\epsilon)w(x,t,\epsilon))\\
& \approx (2v_1^2+2u_1w_1)H(-x+2u_1t,\epsilon)\\
&\quad +(2v_2^2+2u_2w_2)\{1-H(-x+2u_2t,\epsilon)\}\\
&\quad +2u_1e_1(t)\int{w_{0u}(y)w_e(y)dy}\delta(-x+2u_1t,\epsilon)\\
&\quad +2u_2e_2(t)\int {w_{0u}(y)w_e(y)dy}\delta(-x+2u_2t,\epsilon).
\end{aligned} \label{n6}
\end{equation}
Taking the distributional derivative of $w$ with respect to $t$, we obtain
\begin{equation}
\begin{aligned}
&w_t(x,t,\epsilon)\\&\approx\{\dot{e_1}(t)+2u_1w_1\}\delta(-x+2u_1t,\epsilon)\\&+\{\dot{e_2}(t)-2u_2w_2\}\delta(-x+2u_2t,\epsilon)
+2u_1e_1(t)\delta'(-x+2u_1t,\epsilon)\\&+2u_2e_2(t)\delta'(-x+2u_2t,\epsilon).
\label{n7}
\end{aligned}
\end{equation}
Taking the distributional derivative of the expression in \eqref{n6},
 with respect to $x$, we obtain
\begin{equation}
\begin{aligned}
&2(v^2(x,t,\epsilon)+u(x,t,\epsilon)w(x,t,\epsilon))_x\\
 & \approx -(2v_1^2+2u_1w_1)\delta(-x+2u_1t,\epsilon)\\
&\quad +(2v_2^2+2u_2w_2)\delta(-x+2u_2t,\epsilon)\\&
-2u_1e_1(t)\int{w_{0u}(y)w_e(y)dy}\delta'(-x+2u_1t,\epsilon)\\
&\quad -2u_2e_2(t)\int {w_{0u}(y)w_e(y)dy}\delta'(-x+2u_2t,\epsilon).
\end{aligned} \label{n8}
\end{equation}
Using  expansions \eqref{n7} and \eqref{n8} in the third equation of \eqref{e1.2},
 we obtain
\begin{equation}
\begin{aligned}
&w_t(x,t,\epsilon)+2(v^2(x,t,\epsilon)+u(x,t,\epsilon)w(x,t,\epsilon))_x\\
& \approx\{\dot{e_1}(t)-2v_1^2\}\delta(-x+2u_1t)
 + \{\dot{e_2}(t)+2v_2^2\}\delta(-x+2u_2t)\\
&\quad +2u_1e_1(t)\{1-\int {w_{0u}(y)w_e(y)dy}\}\delta'(-x+2u_1t)\\
&\quad + 2u_2e_2(t)\{1-\int {w_{0u}(y)w_e(y)dy}\}\delta'(-x+2u_2t).
\end{aligned} \label{n9}
\end{equation}
So expression \eqref{n9} is zero, if the coefficients of $\delta(-x+2u_1t)$
and $\delta'(-x+2u_1t)$ are zero. Which implies
\begin{equation}
\int w_{0u}(y)w_e(y)dy=1,\quad
 \dot{e_1}(t)=2v_1^2\, \quad \dot{e_2}(t)=-2v_2^2.
\label{nh1}
\end{equation}
Now we find the asymptotic expansion of terms appearing in the fourth
equation of \eqref{e1.2}. Applying lemma \ref{lem2.3},
\begin{equation}
\begin{aligned}
v(x,t,\epsilon)w(x,t,\epsilon)
& \approx v_1w_1 H(-x+2u_1t,\epsilon)+v_2w_2\{1-H(-x+2u_2t,\epsilon)\}\\
&\quad +v_1e_1(t)\int{w_{0v}(y)w_e(y)dy}\delta(-x+2u_1t,\epsilon)\\
&\quad +v_2e_2(t)\int {w_{0v}(y)w_e(y)dy}\delta(-x+2u_2t,\epsilon).
\end{aligned} \label{n10}
\end{equation}


\begin{equation}
\begin{aligned}
&u(x,t,\epsilon)z(x,t,\epsilon)\\
& \approx u_1z_1H(-x+2u_1t,\epsilon)+u_2z_2\{1-H(-x+2u_2t,\epsilon)\}
\\
&\quad+u_1g_1(t)\int{w_{0u}(y)w_g(y)dy}\delta(-x+2u_1t,\epsilon)
\\
&\quad +u_2g_2(t)\int {w_{0u}(y)w_g(y)dy}\delta(-x+2u_2t,\epsilon)
\\
&\quad +u_1h_1(t)\int{w_{0u}(y)w_g(y)dy}\delta'(-x+2u_1t,\epsilon)
\\
&\quad +u_2h_2(t)\int {w_{0u}(y)w_g(y)dy}\delta'(-x+2u_2t,\epsilon)
\\
&\quad +\{u_1P_1(t)\int{w'_{u}(y)w'_P(y)dy}-u_1h_1(t)\int{w_{u}(y)w_g(y)dy}\}
\frac{1}{\epsilon}\delta(-x+2u_1t,\epsilon)
\\
&\quad +\{u_2P_2(t)\int{w'_{u}(y)w'_P(y)dy}-u_2 h_2(t)\int {w_{u}(y)w_g(y)dy}\}
\frac{1}{\epsilon}\delta(-x+2u_2t,\epsilon).
\label{n11}
\end{aligned}
\end{equation}
Using \eqref{n10} and \eqref{n11}, we have
\begin{align}
&3v(x,t,\epsilon)w(x,t,\epsilon)+u(x,t,\epsilon)z(x,t,\epsilon) \nonumber\\
& \approx \{3v_1w_1+u_1z_1\}H(-x+2u_1t,\epsilon)+\{3v_2w_2+u_2z_2\}
 \{1-H(-x+2u_2t,\epsilon)\}
\nonumber \\
&\quad +\{3v_1e_1(t)\int{w_{0v}(y)w_e(y)dy}+u_1g_1(t)\int{w_{0u}(y)w_g(y)dy}\}
 \delta(-x+2u_1t,\epsilon) \nonumber\\
&\quad +\{3v_2e_2(t)\int{w_{0v}(y)w_e(y)dy}+u_2g_2(t)\int{w_{0u}(y)w_g(y)dy}\}
 \delta(-x+2u_2t,\epsilon)
\nonumber\\
&\quad +u_1h_1(t)\int{w_{0u}(y)w_g(y)dy}\delta'(-x+2u_1t,\epsilon)
\nonumber\\
&\quad +u_2h_2(t)\int {w_{0u}(y)w_g(y)dy}\delta'(-x+2u_2t,\epsilon)
\nonumber\\
&\quad +\{u_1P_1(t)\int{w'_{u}(y)w'_P(y)dy}-u_1h_1(t)\int{w_{u}(y)w_g(y)dy}\}
\nonumber \\
&\quad\times \frac{1}{\epsilon}\delta(-x+2u_1t,\epsilon)
\nonumber\\
&\quad+\{u_2P_2(t)\int{w'_{u}(y)w'_P(y)dy}-u_2h_2(t)\int {w_{u}(y)w_g(y)dy}\}
\nonumber\\
&\quad\times \frac{1}{\epsilon}\delta(-x+2u_2t,\epsilon).  \label{n12}
\end{align}
Taking the distributional derivative of the expression in \eqref{n12},
with respect to $x$, we obtain
\begin{equation}
\begin{aligned}
&2((3v(x,t,\epsilon)w(x,t,\epsilon)+u(x,t,\epsilon)z(x,t,\epsilon))_x)
\\
& \approx -2\{3v_1w_1+u_1z_1\}\delta(-x+2u_1t,\epsilon)
+2\{3v_2w_2+u_2z_2\}\delta(-x+2u_2t,\epsilon)
\\
&\quad -2\{3v_1e_1(t)\int{w_{0v}(y)w_e(y)dy}+u_1g_1(t)
 \int{w_{0u}(y)w_g(y)dy}\}\delta'(-x+2u_1t,\epsilon)
\\
&\quad -2\{3v_2e_2(t)\int{w_{0v}(y)w_e(y)dy}+u_2g_2(t)
 \int{w_{0u}(y)w_g(y)dy}\}\delta'(-x+2u_2t,\epsilon)
\\
&\quad -2u_1h_1(t)\int{w_{0u}(y)w_g(y)dy}\delta''(-x+2u_1t,\epsilon)
\\
&\quad -2u_2h_2(t)\int {w_{0u}(y)w_g(y)dy}\delta''(-x+2u_2t,\epsilon)
\\
&\quad -2\{u_1P_1(t)\int{w'_{u}(y)w'_P(y)dy}-u_1h_1(t)\int{w_{u}(y)w_g(y)dy}\}
\frac{1}{\epsilon}\delta'(-x+2u_1t,\epsilon)
\\
&\quad -2\{u_2P_2(t)\int{w'_{u}(y)w'_P(y)dy}-u_2h_2(t)\int {w_{u}(y)w_g(y)dy}\}
\frac{1}{\epsilon}\delta'(-x+2u_2t,\epsilon).
\end{aligned} \label{n13}
\end{equation}
Taking the distributional derivative of $z$  with respect to $t$, we obtain
\begin{equation}
\begin{aligned}
z_t(x,t,\epsilon)
& \approx \{\dot{g_1}(t)+2u_1z_1\}\delta(-x+2u_1t,\epsilon)
\{\dot{g_2}(t)-2u_2z_2\}\delta(-x+2u_2t,\epsilon)\\
&\quad +\{\dot{h_1}(t)+2u_1g_1(t)\}\delta'(-x+2u_1t,\epsilon)\\
&\quad +\{\dot{h_2}(t)+2u_2g_2(t)\}\delta'(-x+2u_2t,\epsilon)\\
&\quad +2u_1h_1(t)\delta''(-x+2u_1t,\epsilon)
+2u_2h_2(t)\delta''(-x+2u_2t,\epsilon).
\end{aligned} \label{n14}
\end{equation}
Using  expansions \eqref{n13} and \eqref{n14} in the fourth
equation of \eqref{e1.2}, we obtain
\begin{align}
&z_t(x,t,\epsilon)+2((3v(x,t,\epsilon)w(x,t,\epsilon)
 +u(x,t,\epsilon)z(x,t,\epsilon))_x) \nonumber \\
& \approx \{\dot{g_1}(t)+2u_1z_1-2(3v_1w_1+u_1z_1)\}\delta(-x+2u_1t,\epsilon)
\nonumber\\
&\quad +\{\dot{g_2}(t)-2u_2z_2+2(3v_2w_2+u_2z_2)\}\delta(-x+2u_2t,\epsilon)
\nonumber\\
&\quad +\{\dot{h_1}(t)2u_1g_1(t)-2(3v_1e_1(t)\int{w_{0v}(y)w_e(y)dy}
\nonumber\\
&\quad +u_1g_1(t)\int{w_{0u}(y)w_g(y)dy})\}\delta'(-x+2u_1t,\epsilon)
\nonumber\\
&\quad +\{\dot{h_2}(t)2u_2g_2(t)-2(3v_2e_2(t)\int{w_{0v}(y)w_e(y)dy}
\nonumber\\
&\quad +u_2g_2(t)\int{w_{0u}(y)w_g(y)dy})\}\delta'(-x+2u_2t,\epsilon)
\nonumber\\
&\quad +2u_1h_1(t)\{1-\int{w_{0u}(y)w_g(y)dy}\}\delta''(-x+2u_1t,\epsilon)
\nonumber\\
&\quad +2u_2h_2(t)\{1-\int{w_{0u}(y)w_g(y)dy}\}\delta''(-x+2u_2t,\epsilon)
\nonumber\\
&\quad -2\{u_1P_1(t)\int{w'_{u}(y)w'_P(y)dy}-u_1h_1(t)\int{w_{u}(y)w_g(y)dy}\}
\nonumber\\
&\quad\times \frac{1}{\epsilon}\delta'(-x+2u_1t,\epsilon)
\nonumber\\
&\quad -2\{u_2P_2(t)\int{w'_{u}(y)w'_P(y)dy}-u_2h_2(t)\int {w_{u}(y)w_g(y)dy}\}
\nonumber\\
&\quad\times \frac{1}{\epsilon}\delta'(-x+2u_2t,\epsilon).
 \label{n15}
\end{align}

Similarly equating the coefficients of
$\delta(-x+2u_it)$, $\delta'(-x+2u_it)$,
 $\delta''(-x+2u_it)$ ,$ \frac{1}{\epsilon}\delta $ and
$\frac{1}{\epsilon} \delta'$, for $i=1,2$ to zero in \eqref{n15}, we obtain
\begin{equation}
\begin{gathered}
\int {w_{0v}(y)w_e(y)dy}=\int {w_{0u}(y)w_g(y)dy}=1,\\
\dot{g_1}(t)=2(3v_1w_1+u_1z_1)-2u_1z_1,\dot{g_2}(t)=2u_2z_2-2(3v_2w_2+u_2z_2),\\
\dot{h_1}(t)=6v_1e_1(t), \quad \dot{h_2}(t)=6v_2e_2(t)\\
\int {w_{u}(y)w_g(y)dy}=1, \quad  \int {w'_{u}(y)w'_P(y)dy}=1\\
 P_1(t)=h_1(t),  \quad  P_2(t)=h_2(t).
\end{gathered} \label{nh2}
\end{equation}
The conditions \eqref{nh1} and \eqref{nh2} together constitute the
 condition \eqref{e2.6}.

To prove it is entropy admissible. Let $\eta$ be a convex
entropy with entropy flux $q$. For this general system it is of the form,
 see \cite{p1}.

\begin{equation}
\begin{gathered}
 \eta(u, v, w, z)=\bar{\eta}(u)+c_1 v+ c_2 w+ c_3 z\\
 q(u,v, w,z)=\bar{q}(u)+c_1 u v+ c_2 (uw+\frac{v^2}{2})+ c_3 (uz+ vw),
\end{gathered}
\end{equation}
where $\bar{\eta}(u)$ is a convex function in the variable $u$ and
satisfies $2 u \bar{\eta}'(u)=\bar{q}'(u)$.
So entropy admissible condition is
\begin{equation*}
  {\limsup_{\epsilon \to 0}}\int_{-\infty}^ \infty
\big[\eta(u^ {\epsilon},v^ {\epsilon}, w^ {\epsilon},z^ {\epsilon} )_t
 + (q (u^ {\epsilon},v^ {\epsilon}, w^ {\epsilon},z^ {\epsilon}))_x\big]
\psi(x)dx\leq 0.
\end{equation*}
for any positive test function $\psi$.
This means
\begin{equation*}
  {\limsup_{\epsilon \to 0}}\int_{-\infty}^ \infty \big[\bar{\eta}(u^ {\epsilon} )_t
 + (\bar{q} (u^ {\epsilon}))_x\big]\psi(x)dx\leq 0.
\end{equation*}
for any test function $\psi\geq 0$.
 We know from the general theory of conservation laws,
 \begin{equation*}
 \begin{aligned}
 & {\limsup_{\epsilon \to 0}}\int_{0}^ \infty
 \int_{-\infty}^ \infty \big[\eta((u^ {\epsilon})_t
 + (q (u^ {\epsilon}))_x\big]\phi(x,t)dx dt\\
 &=\int_{0}^ \infty \int_{-\infty}^ \infty \big[\eta((u)_t
 + (q (u))_x\big]\phi(x,t)dx dt\leq 0,
 \end{aligned}
\end{equation*}
where $u$ is the limit of $u^{\epsilon}$ in the sense of distributions 
and $\phi \in D(\mathbb{R} \times (0, \infty))$.
The last inequality holds because $u$ is an entropy solution
for the conservation law(Burgers equation).

 For any $\epsilon_0 >0$, there exists $\delta >0$ such that for
$0< \epsilon < \delta$, we have
$$
\int_{0}^ \infty \int_{-\infty}^ \infty
\big[\eta((u^\epsilon)_t + (q (u^ \epsilon))_x\big]\phi(x,t)dx dt\leq \epsilon_0
$$
Choose  $\phi(x,t)= \psi(x)\frac{1}{\eta}k(\frac{t-t_0}{\eta})$,
$\int k(t) dt=1$ and $k\geq 0$ is a smooth function. Then
\begin{equation*}
 \int_{0}^ \infty \int_{-\infty}^ \infty
\big[\eta((u^\epsilon)_t + (q (u^ \epsilon))_x \big]\psi(x)
\frac{1}{\eta}k(\frac{t-t_0}{\eta})dx dt\leq \epsilon_0,
 \text{for all } \eta >0.
\end{equation*}
Passing to the limit as  $\eta \to 0$, we have
\begin{equation*}
 \int_{-\infty}^ \infty \big[\eta((u^ \epsilon)_t + (q (u^ \epsilon ))_x\big]
\psi(x)dx dt\leq \epsilon_0\quad \text{for }  t=t_0.
\end{equation*}
Hence \begin{equation*}
  {\limsup_{\epsilon \to 0}}\int_{-\infty}^ \infty \big[\eta((u^ {\epsilon})_t
+ (q (u^ {\epsilon}))_x\big]\psi(x)dx \leq \epsilon_0.
 \end{equation*}
for all $t \geq 0$.
 Since $\epsilon_0$ is arbitrary, we have
$(u^{\epsilon},v^{\epsilon}, w^{\epsilon}, z^{\epsilon})$
is an entropy weak asymptotic solution.

Same analysis as above can be used to prove
$(u^{\epsilon},v^{\epsilon}, w^{\epsilon}, z^{\epsilon})$
is an entropy weak asymptotic solution when $u$ develops shock;
can be seen from the structure of the solution given in
\cite[Theorem 2.3]{hm1}. Hence the proof.
\end{proof}

Piecing together the Riemann problems we construct a weak asymptotic
solution for general type initial data under the assumption that $u$ is a
monotonic decreasing function.

\begin{theorem} \label{thm2.6}
If $u_0,v_0,w_0$ and $z_0$ are bounded measurable functions on
$\mathbb{R}$, and $u_0$ is monotonic, then there exists entropy weak
asymptotic solution $ (u^{\epsilon},v^{\epsilon},w^{\epsilon},z^{\epsilon}) $
to the system \eqref{e1.2} in  $[-K, K]\times [0, \infty)$, for any $K>0$.
\end{theorem}

\begin{proof}
If $u_0$ is a monotonic increasing function, then the result of
 \cite[Theorem 2.5]{hm1} is an entropy solution as can be seen
 easily from the structure given in the \cite[Theorem 2.5]{hm1}.
Lets assume $u_0$ is a monotonic decreasing function. Let $\phi$ be
a test function on $\mathbb{R}$ having support in $[-K, K]$.
Given $\epsilon>0$, there exist piecewise constant functions
$(u_{0\epsilon},v_{0\epsilon},w_{0\epsilon},z_{0\epsilon})$ such that
\begin{equation}
\begin{gathered}
\int_{[-K,K]}|u_0(x)-u_{0\epsilon}(x)|dx
<\epsilon,\quad \int_{[-K,K]}|v_0(x)-v_{0\epsilon}(x)|dx <\epsilon\\
\int_{[-K,K]}|w_0(x)-w_{0\epsilon}(x)|dx
<\epsilon,\quad \int_{[-K,K]}|z_0(x)-z_{0\epsilon}(x)|dx <\epsilon.
\end{gathered}
\end{equation}
In addition to this we can take $u_{0\epsilon}$ monotonic decreasing and
all functions have same points of discontinuities.
 $(u_{0\epsilon},v_{0\epsilon},w_{0\epsilon},z_{0\epsilon})$
in $[-K,K]$ can be represented as
\begin{equation} 
\begin{gathered}
u_{0\epsilon}=\sum_{i=1}^{n}{u_{0i}(H(x-a_{i-1})-H(x-a_{i}))}\\
v_{0\epsilon}=\sum_{i=1}^{n}{v_{0i}(H(x-a_{i-1})-H(x-a_{i}))}\\
w_{0\epsilon}=\sum_{i=1}^{n}{w_{0i}(H(x-a_{i-1})-H(x-a_{i}))}\\
z_{0\epsilon}=\sum_{i=1}^{n}{z_{0i}(H(x-a_{i-1})-H(x-a_{i}))}.
\end{gathered}
\end{equation}
Since $ u_{0\epsilon} $ is a monotonic decreasing function, discontinuity
curve arising in the solution of $ (u,v,w,z)$ do not intersect for any
time. So the following functions are weak asymptotic solutions
\begin{align*}
u(x,t,\epsilon)
&=\sum_{i=1}^{n-1} \Big[u_{0 i}H_u (-x + u_{0i}t+x_i, \eta)\\
&\quad + \frac{x-x_i}{t} (H_u (-x + u_{0{i+1}}t+x_i, \eta)
 -H_u (-x + (u_{0i}t+x_i), \eta))\\
&\quad +u_{0{i+1}}(1-H_u (-x + u_{0{i+1}}t+x_i,\eta))\Big],
\\ 
v(x,t,\epsilon)
&=\sum_{i=1}^{n-1} \Big[v_{0 i}H_u (-x + u_{0i}t+x_i, \eta)
+v_{0{i+1}}(1-H_u (-x + u_{0{i+1}}t+x_i,\eta))\Big],
\\  
w(x,t,\epsilon)
&=\sum_{i=1}^{n-1} \Big[w_{0 i}H_u (-x + u_{0i}t+x_i, \eta)
+w_{0{i+1}}(1-H_u (-x + u_{0{i+1}}t+x_i,\eta)\Big]
\\ 
&\quad+\sum_{i=1}^{n-1}\Big[ e_{1i}(t)\delta_{e}(-x+u_{0i}t,\eta)
 +e_{2i}(t)\delta_{e}(-x+u_{0{i+1}} t,\eta)\Big]\\ 
z(x,t,\epsilon)
&=\sum_{i=1}^{n-1} \Big[z_{0 i}H_u (-x + u_{0i}t+x_i, \eta)
+z_{0{i+1}}(1-H_u (-x + u_{0{i+1}}t+x_i,\eta))\Big]
\\
&\quad+\sum_{i=1}^{n-1}\Big[ g_{1i}(t)\delta_{e}(-x+u_{0i}t,\eta)
 +g_{2i}(t)\delta_{e}(-x+u_{0{i+1}} t,\eta)\Big]\\
&\quad+\sum_{i=1}^{n-1}\Big[ h_{1i}(t)\delta'_{h}(-x+u_{0i}t,\eta)
 +h_{2i}(t)\delta'_{h}(-x+u_{0{i+1}} t,\eta)\Big]\\
&\quad +\sum_{i=1}^{n-1}[R_{1i} (-x+2u_1t,\epsilon)+R_{2i} (-x+2u_2t,\epsilon)]
\end{align*}
where $ e_{1i}$, $e_{2i}$, $g_{1i}$, $g_{2i}$, $h_{1i}$, $h_{2i}$,
$R_{1i}$ and $R_{2i}$  satisfy \eqref{e2.6}
with $ u_1$, $u_2$, $v_1$, $v_2$, $w_1$, $w_2$, $z_1$, $z_2$, $e_1$,
$e_{2}$, $g_1$, $g_2$, $h_1$, $1$, $h_2$,  $R_1$ and $R_2$
replaced by $ u_{i-1}$, $u_i$, $v_{i-1}$, $v_i$, $w_{i-1}$, $w_i$, $z_{i-1}$,
$z_i$, $e_{1i}$, $e_{2i}$, $g_{1i}$, $g_{2i}$, $h_{1i}$, $e_{2i}$, $h_{1i}$,
$h_{2i}$, $R_{1i}$ and $R_{2i}$ respectively.
In fact, given $\epsilon >0$, we have the following estimates.
\begin{gather*}
\big|\int
L_1[u(x,t,\epsilon)]\psi(x)dx\big|=O(\epsilon),\quad
\big| \int L_2[u(x,t,\epsilon),v(x,t,\epsilon)]\psi(x)\big|=O(\epsilon)\\
\big|\int L_3[u(x,t,\epsilon),v(x,t,\eta(\epsilon)),w(x,t,\epsilon)]
\psi(x)dx\big|=O(\epsilon)\\
\big|\int L_4[u(x,t,\epsilon),v(x,t,\eta(\epsilon)),w(x,t,\eta(\epsilon)),
 z(x,t,\epsilon)]\psi(x)dx\big|=O(\epsilon)\\
\big|\int\Big(u(x,0,\epsilon)-u_0 (x)\Big)\psi(x)dx\big|=O(\epsilon),\quad
\big|\int\Big(v(x,0,\epsilon)-v_0(x)\Big)\psi(x)dx\big|=O(\epsilon)\\
\big|\int\Big(w(x,0,\epsilon)-w_0 (x)\Big)\psi(x)dx\big|= O(\epsilon),\quad
\big|\int\Big(z(x,0,\epsilon)-z_0 (x)\Big)\psi(x)dx\big|=O(\epsilon),
\end{gather*}
Therefore $({u},{v},{w},{z})$ is a weak asymptotic solution to system
\eqref{e1.2}--\eqref{e1.3}.
Similar proof as in Theorem \ref{thm2.4} gives $({u},{v},{w},{z})$ is an entropy
weak asymptotic solution.
\end{proof}

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referee whose remarks
helped in improving this article.

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