\documentclass[twoside]{article}
\usepackage{amssymb, amsmath} % font used for R in Real numbers
\pagestyle{myheadings}

\markboth{\hfil $L^1$ singular limit \hfil EJDE--2003/23}
{EJDE--2003/23\hfil Christian Klingenberg, Yun-guang Lu, \&  Hui-jiang Zhao \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2003}(2003), No. 23, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  $L^1$ singular limit for relaxation
and viscosity approximations of extended traffic flow models
 %
\thanks{ {\em Mathematics Subject Classifications:} 35B40, 35L65.
\hfil\break\indent {\em Key words: }  $L^1$ singular limit,
traffic flow model, relaxation and viscosity approximation.
\hfil\break\indent
\copyright 2003 Southwest Texas State University. \hfil\break\indent
Submitted October 25, 2002. Published March 7, 2003.} }
\date{ }
%
\author{Christian Klingenberg, Yun-guang Lu, \&  Hui-jiang Zhao}
\maketitle

\begin{abstract}
 This paper considers the Cauchy problem for an extended 
 traffic flow model with $L^1$-bounded initial data. 
 A solution of the corresponding equilibrium equation
 with $L^1$-bounded initial data is given by the limit of
 solutions of viscous approximations of the original
 system as the dissipation parameter $\epsilon$ tends to zero 
 more slowly than the response time $\tau$. The proof of 
 convergence is obtained by applying the Young measure to 
 solutions introduced by DiPerna and, based on the estimate
 $$
 |\rho(t,x)| \leq \sqrt {|\rho_0(x)|_1/(ct)}
 $$
 derived from one of Lax's results and Diller's idea, 
 the limit function $\rho(t,x)$ is shown to be a $L^1$-entropy 
 week solution. A direct byproduct is that we can get the 
 existence of $L^1$-entropy solutions for the Cauchy problem  
 of the scalar conservation law with $L^1$-bounded initial data
 without any restriction on the growth exponent of the flux 
 function provided that the flux function is strictly convex.  
 Our result shows that, unlike the weak solutions of the 
 incompressible fluid flow equations studied by DiPerna 
 and Majda in \cite{d3}, for convex scalar conservation laws with 
 $L^1$-bounded initial data, the concentration phenomenon will 
 never occur in its global entropy solutions.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}


\section{Introduction }

 In this paper we are concerned with the Cauchy problem
for the extended traffic flow model
\begin{equation}
\begin{gathered}
\rho_{t}+( \rho u)_{x}=0,  \\
u_{t}+\big( \frac{u^{2}}{2}+f( \rho)\big)_{x}+ \frac{h( \rho)(u-c
\rho)}{ \tau}=0,
\end{gathered}
\end{equation}
with $L^1$-bounded initial data
\begin{equation}
( \rho (0,x),u(0,x))=( \rho_0(x),u_0(x))\in L^1(\mathbb{R},\mathbb{R}^2),
\end{equation}
where $c$ is a constant and $ \tau$ denotes the response-time.

The existence of global classical solution of (1.1)  was
obtained by Schochet \cite{s1} for the case
$f( \rho)= \frac{ \mu}{\tau} \log \rho$ under the assumptions that $ \tau$ is
sufficiently small and $ \tau \leq \mu^{3+ \alpha} ( \alpha>0)$.
The zero relaxation limit in the $L^{ \infty}$ setting for
related systems of (1.1) was considered in the recent paper \cite{l5}.

In this paper we show that an $L^1$-solution of the equilibrium
equation
\begin{equation}
\rho_{t}+(c \rho^{2})_{x}=0
\end{equation}
with $L^1$-bounded initial data $\rho_0(x)$ can be obtained by
the limit of viscous solutions of the original system (1.1)
\begin{equation}
\begin{gathered}
\rho_{t}+( \rho u)_{x}= \epsilon \rho_{xx},          \\
u_{t}+\big( \frac{u^{2}}{2}+f( \rho)\big)_{x}
+ \frac{h( \rho)(u-c\rho)}{ \tau}= \epsilon u_{xx},
\end{gathered}
\end{equation}
as the dissipation parameter $ \epsilon$ and the response-time $\tau$
tend to zero, with $ \tau =o( \epsilon)$.

For a large class of functions $f( \rho)$ the solutions of the
parabolic systems (1.4) have no {\it a-priori} $L^{\infty}$-estimates
which are independent of the viscous parameter
$ \epsilon$ even if the initial data are bounded in $L^{ \infty}$ and
sufficiently smooth. Fortunately under suitable restrictions on
the nonlinear functions $f$ and $h$, we can get the following
estimates, in which $| \cdot |_{p}$ denotes the norm on $L^{p}$
and $| \cdot|_{p,q} $ equals $|\cdot|_{p}+|\cdot|_{q}$,
\begin{equation}
\left| \rho^{ \epsilon, \tau,m}(t,x)\right|_{p} \leq M \left|
\rho_0^{m}(x)\right|_{1, \infty}, \quad p >1,
\end{equation}
for solutions $( \rho^{ \epsilon, \tau,m}(t,x), u^{ \epsilon, \tau,m}(t,x))$
of the Cauchy problem (1.4) with initial data
\begin{equation}
( \rho^{ \epsilon, \tau,m}(t,x), u^{ \epsilon, \tau,m}(t,x))|_{t=0} =(
\rho_0^{m}(x),u_0^{m}(x)),
\end{equation}
where $ \rho_0^{m}(x), u_0^{m}(x)$, which are smooth
functions obtained by smoothing the initial data $ ( \rho_0(x),
u_0(x))$ with a mollifier, satisfy
\begin{equation}
\begin{gathered}
( \rho_0^{m}(x), u_0^{m}(x)) \in L^1 \cap L^{ \infty}
 \cap C^{ \infty}(\mathbb{R},\mathbb{R}^{2}),\\
| \rho_0^{m}(x)|_1 \leq | \rho_0(x)|_1, \quad
|u_0^{m}(x)|_1 \leq |u_0(x)|_1,
\end{gathered}
\end{equation}
for any fixed $m > 0$,  and
\begin{equation}
\rho_0^{m}(x) \rightarrow \rho_0(x), \quad
u_0^{m}(x) \rightarrow  u_0(x) \quad \mbox{a.e. in
$L^1$ as $m \rightarrow 0$.}
\end{equation}
When $ \epsilon$ and $\tau$ tend to zero related by $ \tau=o( \epsilon)$, for
any fixed $m > 0$, we can prove that the Young measure $
\nu^m_{(t,x)} $ associated to the sequence $\{ \rho^{ \epsilon,
\tau,m}(t,x)\}$ is an entropy measure valued solution of (1.3).
Then by applying the results in \cite{s2}, $ \nu^m_{(t,x)}$ is a
Dirac measure  and the limit $ \rho^{m}(t,x)$ of $ \rho^{ \epsilon,
\tau,m}(t,x)$ is the unique $L^{\infty}$-entropy solution of the Cauchy
problem $(1.3)$ with the initial data $\rho_0^{m}(x)$.
Furthermore, according to the results obtained in \cite{s2}, we have
that such a solution $\rho^m(t,x)$ can be obtained as the strong
limit of the solution sequence $\{\rho^{\beta,m}(t,x)\}$ to the
following Cauchy problem
\begin{gather*}
\rho_t+\left(c\rho^2\right)_x=\beta \rho_{xx},\\
\rho(t,x)|_{t=0}=\rho_0^m(x),
\end{gather*}
as $\beta\rightarrow 0^+$. Since $\rho^m_0(x)$ satisfies (1.7), we have from
the well-known result of Kruzkov \cite{k2} that
\begin{equation}
\int_{\mathbb{R}} | \rho^{m_1}(t,x)- \rho^{m_{2}}(t,x)| dx
\leq \int_{\mathbb{R}} | \rho_0^{m_1}(x)-
\rho_0^{m_{2}}(x)| dx,
\end{equation}
which means that $ \rho^{m}(t,x)$ is a Cauchy sequence in $L^1$.

Note that the flux function $c \rho^{2}$ in $(1.3)$ is a
strictly convex function, we have from the results obtained by
Lax in \cite{l2} that $ \rho^{m}(t,x)$ is the almost everywhere unique
minimizer of the functional
\begin{equation}
\phi (t,x,v)= \int_{x}^{x+2cv} \rho_0^{m}(x) dx
+cv^{2}t.
\end{equation}
However, since for any fixed point $(t,x)$, $\phi (t,x,0)=0$, and
\begin{equation}
 \phi (t,x,v) \geq - \int_{\mathbb{R}} | \rho^m_0(x)| dx +cv^{2}t >0
\end{equation}
if
$$
|v| \geq \sqrt { \frac{|\rho_0(x)|_1}{ct} }.
$$
Consequently $ \rho^{m}(t,x)$ must satisfy the  estimate
\begin{equation}
| \rho^{m}(t,x)| \leq \sqrt { \frac{| \rho^m_0(x)|_1}{ct}
}\leq \sqrt { \frac{|\rho_0(x)|_1}{ct} }.
\end{equation}
Thus from (1.9) and (1.12), there exists a function $ \rho(t,x)
\in L^1_{\rm loc}(\mathbb{R}^{+} \times \mathbb{R})$ such that $ \rho^{m}(t,x)
\rightarrow \rho(t,x)$ and the limit function $ \rho(t,x)$ is a
$L^1$-entropy solution, in the sense of Szepessy \cite{s2} and Diller
\cite{d1}, of the Cauchy problem $(1.3)$ with $L^1$-initial data $
\rho_0(x)$. Furthermore, following the arguments developed by
Diller in \cite{d1}, the $L^1$-entropy weak solution, which satisfies
the estimate (1.12), to the Cauchy problem (1.3) with
$L^1$-initial data $\rho_0(x)$ is unique and depends continuously
in $L^1$-norm on the initial data and such a uniqueness result
guarantees that the whole sequence of $\{(\rho^{\epsilon,\tau,m}(t,x),
u^{\epsilon,\tau,m} (t,x))\}$ converges strongly to $(\rho(t,x),
u(t,x))$.

In this paper we assume $f,h$ and the initial
data $(\rho_0(x), u_0(x))$ satisfy the following hypotheses:
For $q<4$, $p<8$ and positive constants $c_1\dots c_4$ we have
\begin{itemize}
\item[A1] $\frac{f'( \rho)}{ \rho} \geq c_1> c^{2}$,
$|f'( \rho)| \leq M(1+| \rho|^{q})$;

\item[A2] $c_{2}(1+| \rho|^{4}) \leq h( \rho) \leq c_{3}(1+| \rho|^{p})$;

\item[A3] $ |\rho_0|_1 \leq c_{4}, |u_0|_1 \leq c_{4}$,
\end{itemize}

 \section{Viscous Solutions}

 In this section, we consider global existence
results for the parabolic system (1.4) with initial data
(1.6). Since for any fixed $m>0$, $(\rho_0^{m}(x)$,
$u_0^{m}(x))$ are bounded in $L^{ \infty}$, by applying the
standard contracting map principle to an integral representation
of $(1.4)$, the local existence of $L^{ \infty}$ solutions, for
fixed $ \epsilon, \tau$, to the Cauchy problem (1.4), (1.6) can be
easily established.

To extend a local solution to a global solution, we use the the following
{\it a-priori} $L^{ \infty}$-estimates.

\begin{lemma} \label{lm1}
If $h( \rho)$ and $f( \rho)$ satisfy the assumptions A1, A2,
 $( \rho_0^{m}(x),u_0^{m}(x))$ satisfies (1.7) and the smooth
 solutions
$( \rho^{ \epsilon, \tau,m}(t,x),u^{ \epsilon, \tau,m}(t,x))$ of the Cauchy
problem (1.4), (1.6) exist in $[0,T] \times \mathbb{R}$,  then the
following estimates hold
\begin{equation}
| \rho^{ \epsilon, \tau,m}(t,x)| \leq C(T, \epsilon, \tau,m), \quad
|u^{ \epsilon, \tau,m}(t,x)| \leq C(T, \epsilon, \tau,m),
\end{equation}
where $C(T, \epsilon, \tau,m)$ is a positive constant depending on $ T,
\epsilon, \tau$ and $m$.
Furthermore if $ M_1 \tau \leq  \epsilon$ for a suitable large
constant $M_1$, then
\begin{equation}
| \rho^{ \epsilon, \tau,m}(t,x)|_{3} \leq M(m), \quad
|\rho^{\epsilon,\tau,m}(t,x)|_1 \leq |\rho_0(x)|_1,
\end{equation}
where $M(m)$ is a positive constant independent of $\epsilon$ and $\tau$, but
depending on $m$.
\end{lemma}

\paragraph{Proof}
For simplicity, we omit superscripts in $(\rho^{ \epsilon,\tau,m}(t,x),
u^{ \epsilon, \tau,m}(t,x))$.
Multiplying the first equation in (1.4) by
$ \int_0^{ \rho} \frac{f'(s)}{s} ds$
 and the second equation by $u-c \rho$, then
adding the results and integrating on $[0,t] \times \mathbb{R}$, we
have
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}}\Big( \frac{1}{2}u^{2}+
\int_0^{ \rho}
\int_0^{y}\frac{f'(s)}{s}\, ds\,dy-c \rho u\Big) dx+
\int_0^{t} \int_{\mathbb{R}}
\frac{h( \rho)(u-c \rho)^{2}}{ \tau}\,dx\,dt   \\
& +\epsilon \int_0^{t} \int_{\mathbb{R}}
\Big(u_{x}^{2}+2c \rho _{x} u_{x}+ \frac{f'( \rho)}{ \rho}
\rho _{x}^{2}\Big) \,dx\,dt\\
&\leq \int_{\mathbb{R}} \Big(\frac{1}{2}\left
(u_0^{m}\right)^{2}+ \int_0^{ \rho_0^{m}}
\int_0^{y}\frac{ f'(s)}{s} dsdy-c \rho_0^{m}
u_0^{m}\Big)dx.
\end{aligned}
\end{equation}
Multiplying the first equation in (1.4) by $ - \rho_{xx}$ and
then integrating on $[0,t]\times \mathbb{R}$, we have
\begin{equation}\begin{aligned}
&\int_{\mathbb{R}} \frac{1}{2} \rho_{x}^{2} dx + \epsilon
\int_0^{t} \int_{\mathbb{R}} \rho_{xx}^{2}\,dx\,dt\\
&= \int_{\mathbb{R}} \frac{1}{2}( \rho_{0x}^{m})^{2} dx+
\int_0^{t}\int_{\mathbb{R}} \rho_{xx}( \rho u)_{x} \,dx\,dt   \\
&\leq  \int_{\mathbb{R}} \frac{1}{2}( \rho_{0x}^{m})^{2}
dx+ \frac{ \epsilon}{2} \int_0^{t} \int_{\mathbb{R}}
(\rho_{xx})^{2}\,dx\,dt + \frac{2}{ \epsilon} \int_0^{t}
\int_{\mathbb{R}} |( \rho u)_{x}|^{2} \,dx\,dt    \\
&\leq  \int_{\mathbb{R}} \frac{1}{2}( \rho_{0x}^{m})^{2}dx
+ \frac{ \epsilon}{2} \int_0^{t} \int_{\mathbb{R}}
(\rho_{xx})^{2}\,dx\,dt + \frac{1}{ \epsilon} | \rho|_{ \infty}^{2}
\int_0^{t} \int_{\mathbb{R}}(u_{x})^{2} \,dx\,dt\\
&\quad + \frac{1}{ \epsilon} |u|_{ \infty}^{2}
\int_0^{t} \int_{\mathbb{R}}( \rho_{x})^{2}\,dx\,dt\,.
\end{aligned}
\end{equation}
Therefore, by (2.3) and A1,
\begin{equation}
\int_{\mathbb{R}} ( \rho_{x})^{2} dx + \epsilon
\int_0^{t} \int_{\mathbb{R}} ( \rho_{xx})^{2}
\,dx\,dt \leq \int_{\mathbb{R}} ( \rho_{0x}^{m})^{2} dx+ C(
\epsilon)\left (| \rho|_{ \infty}^{2}+|u|_{ \infty}^{2}\right).
\end{equation}
Since
\begin{equation}
| \rho|^{2}=2 \int_{- \infty}^{x} \rho \rho_{x} dx
\leq | \rho|_{2} | \rho_{x}|_{2} \leq C( \epsilon)(1+ | \rho|_{
\infty}+|u|_{ \infty}),
\end{equation}
we have
\begin{equation}
| \rho|_{ \infty} \leq C( \epsilon)\big(1+ |u|_{ \infty}^{1/2}\big).
\end{equation}
Multiplying  the second equation in (1.4) by $-u_{xx}$ and then
integrating, we have
\begin{equation}
\begin{aligned}
&\int_{\mathbb{R}} \frac{1}{2}u_{x}^{2} dx+ \epsilon
\int_0^{t}\int_{\mathbb{R}} u_{xx}^{2} \,dx\,dt \\
&=\int_{\mathbb{R}} \frac{1}{2}(u_{0x}^{m})^{2} dx+
\int_0^{t} \int_{\mathbb{R}}
\frac{h( \rho)(u-c \rho)u_{xx}}{ \tau} \,dx\,dt  \\
&\quad + \int_0^{t} \int_{\mathbb{R}} u
u_{x}u_{xx}\,dx\,dt+ \int_0^{t}
\int_{\mathbb{R}}f'( \rho) \rho_{x}u_{xx}\,dx\,dt.
\end{aligned}
\end{equation}
Due to (2.3) and A2,
\begin{equation}\begin{aligned}
&\int_0^{t} \int_{\mathbb{R}} \frac{h( \rho)(u-c\rho)u_{xx}}
{ \tau} \,dx\,dt   \\
&\leq \frac{ \epsilon}{4} \int_0^{t} \int_{\mathbb{R}}u_{xx}^{2} \,dx\,dt+
 \frac{1}{ \epsilon \tau^{2}} \int_0^{t} \int_{\mathbb{R}}
 |h( \rho)|^{2}(u-c \rho)^{2}\,dx\,dt \\
&\leq \frac{ \epsilon}{4} \int_0^{t} \int_{\mathbb{R}}u_{xx}^{2} \,dx\,dt+
\frac{C}{ \epsilon \tau}|h( \rho)|_{ \infty}   \\
&\leq \frac{ \epsilon}{4} \int_0^{t} \int_{\mathbb{R}}u_{xx}^{2} \,dx\,dt+
C( \epsilon, \tau)\left(1+| \rho|_{ \infty}^{p}\right)   \\
&\leq \frac{ \epsilon}{4} \int_0^{t}
\int_{\mathbb{R}}u_{xx}^{2} \,dx\,dt+ C( \epsilon, \tau)\left(1+|u|_{
\infty}^{ \frac{p}{2}}\right),
\end{aligned}
\end{equation}
\begin{equation}
\Big| \int_0^{t} \int_{\mathbb{R}}
uu_{x}u_{xx}\,dx\,dt\Big| \leq \frac{ \epsilon}{4} \int_0^{t}
\int_{\mathbb{R}}u_{xx}^{2} \,dx\,dt+ C( \epsilon, \tau)\left(1+|u|_{
\infty}^{2}\right)
\end{equation}
and
\begin{equation}\begin{aligned}
\Big| \int_0^{t} \int_{\mathbb{R}} f'( \rho)\rho_{x}u_{xx}\,dx\,dt\Big|
& \leq \frac{ \epsilon}{4} \int_0^{t}
\int_{\mathbb{R}}u_{xx}^{2} \,dx\,dt+C( \epsilon)|f'( \rho)|_{ \infty}^{2}   \\
& \leq  \frac{ \epsilon}{4} \int_0^{t}
\int_{\mathbb{R}}u_{xx}^{2} \,dx\,dt+ C( \epsilon)\left(1+|u|_{\infty}^{q}\right),
\end{aligned}
\end{equation}
we have
\begin{equation}
 \int_{\mathbb{R}}u_{x}^{2} dx + \epsilon \int_0^{t} \int_{\mathbb{R}} u_{xx}^{2} \,dx\,dt
\leq \int_{\mathbb{R}}(u_{0x}^{m})^{2} dx+C( \epsilon,
\tau)\left(1+|u|_{ \infty}^{2}+ |u|_{ \infty}^{ \frac{p}{2}}+|u|_{
\infty}^{q}\right).
\end{equation}
Since
\begin{equation}
u^{2} \leq |u|_{2}|u_{x}|_{2} \leq C( \epsilon, \tau) \left(1+|u|_{
\infty}^{2}+ |u|_{ \infty}^{ \frac{p}{2}}+|u|_{ \infty}^{q}\right)^{1/2}
\end{equation}
and $p<8$, $q<4$,
we have by (2.3) and $A1$,
$$|u|_{ \infty} \leq C_1( \epsilon, \tau) < \infty.
$$
Combining the above result with (2.7), we conclude
$| \rho|_{ \infty} \leq C_1( \epsilon, \tau)$.
Therefore, (2.1) is proved.

Now we prove (2.2).
Multiplying  the first equation in (1.4) by $| \rho| \rho $ and
then integrating, we have
\begin{equation}\begin{aligned}
&\frac{1}{3} \int_{\mathbb{R}} | \rho|^{3} dx +
 \epsilon \int_0^{t} \int_{\mathbb{R}} | \rho|
 \rho_{x}^{2} \,dx\,dt  \\
&=\frac{1}{3} \int_{\mathbb{R}} | \rho_0^{m}|^{3} dx + \int_0^{t} \int_{\mathbb{R}} \rho u
(| \rho| \rho)_{x} dx dt  \\
&=\frac{1}{3} \int_{\mathbb{R}} | \rho_0^{m}|^{3} dx +
\int_0^{t} \int_{\mathbb{R}}
\rho (u- c \rho)(| \rho| \rho)_{x} \,dx\, dt
+ c \int_0^{t} \int_{\mathbb{R}} ( \rho)^{2} (| \rho| \rho)_{x} dx dt   \\
&= \frac{1}{3} \int_{\mathbb{R}} | \rho_0^{m}|^{3} dx + \int_0^{t} \int_{\mathbb{R}}
2| \rho| \rho(u- c \rho) \rho_{x} dx dt  \\
&\leq  \frac{1}{3} \int_{\mathbb{R}} | \rho_0^{m}|^{3} dx
+ \tau \int_0^{t} \int_{\mathbb{R}}
\rho_{x}^{2} dx dt+ \int_0^{t}
\int_{\mathbb{R}}
\frac{ \rho^{4} (u- c \rho)^{2}}{ \tau}\,dx\,dt \\
&\leq  M\left(|\rho_0^{m}|_{1, \infty}\right)\big(1+
\frac{ \tau}{ \epsilon}\big).
\end{aligned}
\end{equation}
So the first estimate in (2.2) is proved. Similarly we can prove
the second estimate which completes the proof of Lemma \ref{lm1}.
\hfill$\diamondsuit$

From the {\it a-priori} $ L^{ \infty}$ estimates (2.1) we can
extend the local solution step by step and get the following
global existence theorem.

\begin{theorem} \label{thm2}
If $h( \rho), f( \rho)$ and the initial data satisfy the
conditions A1, A2, and A3, then for any fixed $ \epsilon,
\tau, m$ satisfying $ \tau =o( \epsilon)$, the Cauchy problem (1.4), (1.6)
admits a unique, global smooth solution $( \rho^{\epsilon, \tau,m}(t,x)$,
 $u^{\epsilon, \tau,m}(t,x))$ which
satisfies the estimates (2.1), (2.2).
\end{theorem}

\section{Zero Relaxation and Dissipation Limit}

In this section, we consider the convergence of
solutions $( \rho^{\epsilon, \tau,m}(t,x), u^{\epsilon, \tau,m}(t,x))$ to the
Cauchy problem (1.4), (1.6) as the dissipation parameter $ \epsilon$
and the response time $ \tau$ tend to zero. We show that a
$L^1$-solution of (1.3) with $L^1$-bounded initial data
$\rho_0(x)$ can be given by the limit of
 $ \rho^{ \epsilon, \tau,m}(t,x)$ as $\epsilon+\tau+m\rightarrow 0+$. The technique to
 prove the strong convergence
is to employ the concept of entropy measure valued solution to
(1.3) with initial data $\rho^m_0(x)$ introduced by DiPerna \cite{d2}.

We show that the Young measure $ \nu^m_{(t,x)}$ associated with
$\{ \rho^{ \epsilon, \tau,m}(t,x) \}$ is an entropy measure
valued solution of (1.3) with initial data $\rho_0^m(x)$. Then
by applying the results given in \cite{s2}, we get that
$\nu^m_{(t,x)}$ is a Dirac measure and the limit function
$\rho^{m}(t,x)$ of $\{ \rho^{ \epsilon, \tau, m}(t,x) \}$ as $ \epsilon, \tau$
tend to zero related by $ \tau=o( \epsilon)$ is the unique
$L^{3}$-entropy solution of $(1.3)$ with the initial data
$\rho_0^{m}(x)$. To prove that the limit function $ \rho(t,x)$
of $ \rho^{m}(t,x)$ as $m$ tends to zero is a $L^1$-solution of
(1.3) with the initial data $\rho_0(x)$, we need the
following results of Lax

\begin{lemma}[\cite{l2}] \label{lm3}
 Let $u(t,x)$ be the entropy
solution of the Cauchy
problem for the scalar conservation law
\begin{equation}
\begin{gathered}
u_{t}+(cu^2)_{x}=0,  \\
u(t,x)|_{t=0}=u_0(x) \in L^1\cap L^\infty.
\end{gathered}
\end{equation}
obtained by Kruzkov in \cite{k2}. Then $u(t,x)$
is the almost everywhere unique minimizer of the functional
\begin{equation}
\phi (t,x,v)= \int_{x}^{x+2cv} u_0(x) dx +cv^{2}t.
\end{equation}
\end{lemma}

If  $\rho^m(t,x)$ is the $L^1\cap L^\infty$-entropy solution
for the equation (1.3) with $L^1\cap L^\infty$ initial data
$\rho_0^m(x)$, then from Lemma \ref{lm3} and Diller's result in \cite{d1},
we have
\begin{equation}
|\rho^m(t,x)| \leq \sqrt { \frac{|\rho_0^m(x)|_1}{ct} } \leq
\sqrt { \frac{|\rho_0(x)|_1}{ct} }\,.
\end{equation}
In fact, we have that for any fixed point $(t,x)$,
$ \phi (t,x,0)=0$, and
 \[  \phi (t,x,v) \geq - \int_{\mathbb{R}} |\rho^m_0(x)| dx +cv^{2}t >0  \]
if
$|v| \geq \sqrt {|\rho^m_0(x)|_1/(ct) }$,
and as an immediate consequence, we have that (3.3) holds
almost everywhere.

\begin{theorem} \label{thm4}
The Young measure $ \nu^m_{(t,x)}$ associated to the sequence $ \{ \rho
^{ \epsilon, \tau, m}(t,x) \} $ is an entropic measure valued solution
of the Cauchy problem (1.3) with the initial data
$\rho_0^{m}(x)$.
\end{theorem}

\paragraph{Proof} It is sufficient to prove the following two
estimates \cite{s2}:
\begin{equation}
\frac{\partial}{\partial t}\langle \nu_{(t,x)}^m( \lambda),|\lambda-k|\rangle
 +\frac{ \partial}{ \partial x}\langle\nu_{(t,x)}^m( \lambda),
 \mathop{\rm sign}( \lambda-k) (c \lambda^2-ck^2) \rangle \leq 0
\end{equation}
for all $k \in \mathbb{R}^1$ in the sense of distributions, and
\begin{equation}
\lim_{T \rightarrow 0^{+}} \frac{1}{T} \int_0^{T}
\int_{I} \langle \nu_{(t,x)}^m( \lambda ), |
\lambda-v_0(x)| \rangle \,dx\,dt=0
\end{equation}
for any compact interval $I \in \mathbb{R} $. Since the
function $| \rho -k|$ can be approximated by smooth bounded
convex functions $ \eta( \rho)$ whose first and second
derivatives are bounded in $\mathbb{R}$, the following inequality with the
Young measure representing weak limit theorem \cite{d2,s2} will
give the proof of (3.4):
\begin{equation}
\eta( \rho^{ \epsilon, \tau,m})_{t}+q( \rho^{ \epsilon, \tau,m})_{x} \leq 0
\end{equation}
in the sense of distributions, where $q( \rho)$ is a entropy flux
of (1.3) corresponding to $ \eta ( \rho)$. For brevity, we
will omit the superscripts $\epsilon,\tau$ and $m$ in the following.

To prove (3.6), multiplying  the first
equation in (1.4) by $ \eta'( \rho)$, we have
 \begin{equation}\begin{aligned}
&\eta( \rho)_{t}+q( \rho)_{x}\\
&=- \eta'( \rho)( \rho(u-c  \rho))_{x}+ \epsilon \eta'( \rho) \rho_{xx} \\
&=-( \eta'( \rho) \rho(u-c  \rho))_{x}+ \eta''( \rho)
\rho (u-c  \rho) \rho _{x} - \epsilon \eta''(\rho) \rho _{x}^{2}+ \epsilon
\eta_{xx}(\rho).
\end{aligned}
\end{equation}
 From estimates in (2.4) and $ \tau =o( \epsilon)$, we have that
\begin{equation}
\begin{aligned}
&\int \int_{ \Omega} | \rho
\eta''(\rho)(u-c \rho) \rho_{x}|\,dx\,dt \\
&\leq M\Big(\int \int_{ \Omega} \frac{h( \rho)(u-c \rho)^{2}}{ \tau}
\,dx\,dt\Big)^{1/2}
\Big( \int \int_{ \Omega}
\frac{\tau \rho^{2} \rho_{x}^{2}}{h( \rho)}\,dx\,dt\Big)^{1/2}
\rightarrow 0,
\end{aligned}
\end{equation}
and
\begin{equation}\begin{aligned}
&\big| \int \int_{ \Omega} (\eta'(\rho) \rho
(u-c \rho ))_{x} \phi \,dx\,dt\big|   \\
&= \big| \int \int_{ \Omega}\eta'(\rho)
\rho (u-c \rho ) \phi_{x} \,dx\,dt\big| \\
&\leq  M\Big( \int \int_{ \Omega}\frac{\tau \rho^{2} \phi_{x}^{2}}
{h( \rho)} \,dx\,dt\Big)^{1/2} \Big( \int \int_{ \Omega}
\frac{h( \rho)(u- c \rho)^{2}}{ \tau} \,dx\,dt\Big)^{1/2}
\rightarrow 0
\end{aligned}
\end{equation}
as $ \tau = o( \epsilon)$ and $ \epsilon $  tends to zero for any compact set
$ \Omega $ in $ \mathbb{R} \times \mathbb{R}^{+}$.
Moreover since $ \eta''( \rho) \geq 0$, and $ \epsilon \eta (
\rho)_{xx} \rightarrow 0 $ in the sense of distributions, then
$(3.6)$ is proved by letting $ \epsilon \rightarrow 0 $ in (3.7).

The proof of (3.5) can be obtained as in \cite{s2}; thus
we omit the details.  This completes the proof of Theorem \ref{thm4}.
\hfill$\diamondsuit$

Now we give the main result in this section.

\begin{theorem} \label{thm5}
If $h( \rho), f( \rho)$ and the initial data satisfy the
conditions A1-A3, then the whole solution
sequence of $( \rho^{ \epsilon, \tau, m}(t,x), u^{ \epsilon, \tau, m}(t,x))$
to the Cauchy problem (1.4), (1.6) converges pointwise almost
everywhere
\[
( \rho^{ \epsilon, \tau, m}(t,x), u^{ \epsilon, \tau, m}(t,x)) \rightarrow
( \rho(t,x), u(t,x))
\]
as $m, \epsilon$ and $ \tau$ tend to zero whose
relation are given by $ \tau=o( \epsilon)$. Here the limit functions
$(\rho(t,x), u(t,x))$ satisfy
\begin{enumerate}
\item $u(t,x)=c \rho (t,x)$ for almost all $(t,x) \in \mathbb{R}^+ \times
\mathbb{R}$ and

\item $ \rho(t,x)$ is the unique $L^1$-entropy solution of the
Cauchy problem
(1.3) with $L^1$-bounded initial data $ \rho_0(x)$, which
satisfies the estimate (3.3).
\end{enumerate}
\end{theorem}

\paragraph{Proof}  From Theorem \ref{thm4} and the results obtained in
\cite{s2}, we
conclude that
$$
\nu^m_{(t,x)}=\delta_{\rho^m(t,x)},\quad \mbox{a. e.,}
$$
 From Lemma \ref{lm1}, there exists a subsequence
$\{\rho^{\epsilon_k,\tau_k,m}(t,x)\}$ of $\{\rho^{\epsilon,\tau,m}(t,x)\}$ such that
$$
\rho^{\epsilon_k,\tau_k,m}(t,x)\rightarrow \rho^m(t,x)\quad\mbox{in }
L^1(\mathbb{R}^+\times \mathbb{R}) \quad\mbox{as }
\epsilon_k+\tau_k\rightarrow 0^+ \eqno(3.10)
$$
provided $\tau_k=o(\epsilon_k)$. One can easily verify that
$\rho^m(t,x)$ is a $L^{\infty}$-entropy week solution, in the sense of
\cite{s2}, to (1.3) with initial data $\rho^m_0(x)$.

On the other hand, (2.3) and A2 imply
$$
\lim_{\epsilon_k+\tau_k\rightarrow 0+}
\int^t_0\int_\mathbb{R}\left|u^{\epsilon_k,\tau_k,m}(t,x)-c\rho^{\epsilon_k,\tau_k,m}
(t,x)\right| \,dx\,dt=0
\eqno(3.11)
$$
which means that there exists a function $u^m(t,x)=c\rho^m(t,x)$ such that
$$
u^{\epsilon_k,\tau_k,m}(t,x)\rightarrow u^m(t,x)\quad\mbox{in }
L^1(\mathbb{R}^+\times \mathbb{R}) \quad\mbox{as }
\epsilon_k+\tau_k\rightarrow 0^+.
\eqno(3.12)
$$
Furthermore, by employing the results obtained in
\cite{s2} again, we have that
the solution $\rho^m(t,x)$ obtained above can be obtained as the strong limit
of the solution sequence $\{\rho^{\beta,m}(t,x)\}$ to the following Cauchy problem
\begin{gather*}
\rho_t+\left(c\rho^2\right)_x=\beta \rho_{xx},\\
\rho(t,x)|_{t=0}=\rho_0^m(x),
\end{gather*}
as $\beta\rightarrow 0^+$. Since $\rho^m_0(x)\in L^1\cap L^\infty$,
we have from the well-known result of Kruzkov \cite{k2} that
$$
\int_{\mathbb{R}} | \rho^{m_{2}}(t,x)- \rho^{m_1}(t,x)| dx
\leq \int_{\mathbb{R}} | \rho_0^{m_{2}}(x)-
\rho_0^{m_1}(x)| dx, \eqno(3.13)
$$
and from the discussions after Lemma \ref{lm3}, $\rho^m(t,x)$ must satisfy the
estimate (3.3). Consequently
$$
\rho^m(t,x)\rightarrow \rho(t,x) \quad\mbox{in }
L^1(\mathbb{R}^+\times \mathbb{R}),
\eqno(3.14)
$$
and $\rho(t,x)$ satisfies
$$
|\rho(t,x)|\leq \sqrt{\frac{|\rho_0(x)|_1}{ct}}.
\eqno(3.15)
$$
Diller's results in \cite{d1}  show that $\rho(t,x)$ is a $L^1$-entropy
weak solution of (1.3) with $L^1$-bounded initial data $\rho_0(x)$ in
the sense
of Kruzkov \cite{k2}. So the first assertion of Theorem \ref{thm5} is easy
to be verified by (3.11), (3.12) and (1.8).

To conclude that the whole sequence of
$\{(\rho^{\epsilon,\tau,m}(t,x), u^{\epsilon,\tau,m}(t,x))\}$ converges almost
everywhere to $(\rho(t,x), u(t,x))$, we need only to
prove the uniqueness of the $L^1$-entropy weak solution $\rho(t,x)$,
which satisfies
(3.15), to the Cauchy problem (1.3) with $L^1$-bounded initial data
$\rho_0(x)$. Due to the estimate (3.15), such a result follows from
the same arguments developed by
Diller in \cite{d1}. This completes the proof of Theorem \ref{thm5}.

\paragraph{Acknowledgments}
The authors are grateful to the referees for their carefully reading
the original manuscript and for their valuable suggestions.
The authors were partially supported by a grant from National University
of Colombia, by grants  10071080 and 10041003 of NNSF China.

\begin{thebibliography}{99} \frenchspacing

\bibitem{c1} I.-L. Chern, Long-time effect of relaxation for hyperbolic
conservation laws,  {\it Comm. Math. Phys.} {\bf 172} (1995),
39-55.

\bibitem {c2} G.-Q. Chen, C.D. Levermore, T.-P. Liu, Hyperbolic conservation laws
with stiff relaxation terms and entropy, {\it Comm. Pure Appl.
Math.} {\bf 47} (1994), 787-830.

\bibitem {c3} G.-Q. Chen, T.-P. Liu, Zero relaxation and dissipation limits for
hyperbolic conservation laws, {\it Comm. Pure Appl. Math.} {\bf
46} (1993), 755-781.

\bibitem {d1} D. J. Diller, A note on the uniqueness of entropy solution to
first order quasilinear equations, {\it Electronic J. of
Differential Equations} {\bf 5} (1994), 1-4.

\bibitem {d2} R. J. DiPerna, Measure-valued solutions to conservation laws,
{\it Arch. Rational Mech. Anal.} {\bf 88} (1995), 223-270.

\bibitem {d3} R. J. DiPerna and A. Majda, Oscillations and concentrations
in weak solutions of the incompressible fluid equations, {\it Comm.
Math. Phys.} {\bf 108} (1987), 667-689.

\bibitem {k1} C. Klingenberg, Y.-G. Lu, Cauchy problem for hyperbolic
conservation laws with a relaxation term, {\it Proc. Roy. Soc. of
Edinb.} {\bf 126A} (1996), 821-828.

\bibitem {k2} S. N. Kruzkov, First order quasilinear equations in several
independent variables, {\it Mat. SB.(N.S.)} {\bf 81} (1970),
228-255.

\bibitem {l1} C. Lattanzio and P. Marcati, The zero relaxation limit for the
hydrodynamic Whitham traffic flow model, {\it J.
Differential Equations} {\bf 141} (1997), 150-178.

\bibitem {l2} P. D. Lax, {\it Hyperbolic systems of conservation laws and the
mathematical theory of shock waves}, SIAM, Philadelphia, 1973.

\bibitem {l3} Y.-G. Lu and C. Klingenberg, The Cauchy problem for hyperbolic
consedvation laws with three equations, {\it J. Math. Appl.
Anal.} {\bf 202} (1996), 206-216.

\bibitem {l4} Y.-G. Lu and C. Klingenberg, The relaxation limits for systems of
 Broadwell type, {\it  Differential and Integral Equations} {\bf 14}
(2001), 117-127.

\bibitem {l5} Y.-G. Lu, Singular Limits of Stiff Relaxation and Dominant Diffusion
   for Nonlinear Systems,  {\it  J. Differential Equations} {\bf 179} (2002), 687-713.

\bibitem {l6} Y.-G. Lu, {\it Hyperboilc Conservation Laws and the Compensated
   Compactness Method}, CRC Press, USA, 2002.

\bibitem {s1} S. Schochet, The instant-response limit in Whitham's nonlinear
traffic-flow model: Uniform well-posedness and global existence,
{\it Asymptotic Analysis} {\bf 1} (1988), 4: 263-282.

\bibitem {s2} A. Szepessy, An existence result for scalar conservation laws
using measure valued solutions, {\it Commun. Partial Differential
Equations} {\bf 14} (1989),  1329-1350.

\bibitem {w1} G. B. Whitham, {\it Linear and Nonlinear Waves},
John Wiley and Sons, 1973.

\bibitem {z1} Huijiang Zhao, A note on the Cauchy problem to a class of
nonlinear dispersive equations with singular initial data, {\it
Nonlinear Analysis, TMA} {\bf 42} (2000), 251-270.

\end{thebibliography}

\noindent\textsc{Christian Klingenberg} \\
Department of Mathematicas, W\"urzburg University\\
W\"urzburg, 97074, Germany \\
 e-mail:  klingen@mathematik.uni-wuerzburg.de\smallskip

\noindent\textsc{Yun-guang Lu} \\
Department of Mathematics \\
University of Science and Technology of China, Hefei, China\\
 and \\
Departamento de Matem\'aticas\\
Universidad Nacional de Colombia, Bogota, Colombia \\
e-mail:  yglu@matematicas.unal.edu.co \smallskip

\noindent\textsc{Hui-jiang Zhao} \\
Institute of Physics and Mathematicas \\
Chinese Academy of Sciences, Wuhan, China \\
e-mail: hjzhao@wipm.whcnc.ac.cn

\end{document}