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\markboth{\hfil $L^1$ stability of conservation laws \hfil EJDE--2001/14}%
{EJDE--2001/14\hfil Tong Li \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf2001}(2001), No. 14, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
$L^1$ stability of conservation laws \\ for a traffic flow model
%
\thanks{ {\em Mathematics Subject Classifications:}
35L65, 35B40, 35B50, 76L05, 76J10.
\hfil\break\indent
{\em Key words:} Relaxation, shock, rarefaction, $L^1$-contraction,
traffic flows, anisotropic, \hfil\break\indent
equilibrium, marginally stable, zero relaxation limit, large-time behavior, $L^1$-stability.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted June 2, 2000. Published February 20, 2001.} }
\date{}
\author{Tong Li}
\maketitle
\begin{abstract}
We establish the $L^1$ well-posedness theory
for a system of nonlinear hyperbolic
conservation laws with relaxation arising in traffic flows.
In particular, we obtain the continuous dependence
of the solution on its initial data in $L^1$ topology.
We construct a functional
for two solutions which is equivalent to the $L^1$ distance
between the solutions. We prove that the functional
decreases in time which yields the $L^1$ well-posedness of the Cauchy
problem.
We thus obtain the $L^1$-convergence to and the uniqueness of
the zero relaxation limit.
We then study the large-time behavior of the entropy solutions.
We show that the equilibrium shock waves are nonlinearly stable in $L^1$
norm. That is, the entropy solution with initial data as certain $L^1$-bounded
perturbations of an equilibrium shock wave exists globally and
tends to a shifted equilibrium shock wave in $L^1$ norm
as $t\to \infty$.
We also show that if the initial data $\rho_0$ is bounded and of compact support,
the entropy solution converges in $L^1$ to an equilibrium $N$-wave as
$t\to +\infty$.
\end{abstract}
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\section{Introduction}
We establish the $L^1$ well-posedness theory
for a system of nonlinear hyperbolic
conservation laws with relaxation arising in traffic flows.
In particular, we obtain the continuous dependence
of the solution on its initial data in $L^1$ topology,
the $L^1$-convergence to and the uniqueness of
the zero relaxation limit.
We then show that the equilibrium shock waves
are nonlinearly stable in $L^1$ norm.
The $L^1$ topology is natural from point view of the conservation laws.
The well-posedness problem in the $L^1$ topology
for nonlinear conservation laws has been
studied, see Bressan, Liu and Yang \cite{BLY},
Liu and Yang \cite{LiuY}.
$L^1$-stability of shock waves in scalar conservation laws
has been studied, see Freist\"uhler and Serre \cite{FS},
Mascia and Natalini \cite{Mana}, Natalini \cite{Nata}.
The system of nonlinear hyperbolic conservation laws with relaxation
we study was derived
as a nonequilibrium continuum model of traffic flows by Zhang \cite{Zhang},
also see Li and Zhang \cite{LiZ}.
The main purpose of the model is
to address the anisotropic feature of traffic flows.
The resulting
hyperbolic system with relaxation
is marginally stable.
The model is the following
\begin{eqnarray}
\rho_t+(\rho v)_x&=&0\label{5}\\
v_t+(\frac{1}{2}v^2+g(\rho))_x&=&\frac{v_e(\rho)-v}{\tau}\label{6}
\end{eqnarray}
with initial data
\begin{eqnarray}
(\rho(x,0), v(x,0))=(\rho_0(x), v_0(x)).\label{i2}
\end{eqnarray}
It is assumed that
\begin{eqnarray}
\rho_0(x)\geq\delta_0>0\label{assu1}
\end{eqnarray}
for some $\delta_0>0$. %$v_e(\rho)$ is the equilibrium velocity.
$g$ is the anticipation factor satisfying
\begin{eqnarray}
g'(\rho)=\rho(v'_e(\rho))^2.\label{G}
\end{eqnarray}
$\tau>0$ is the relaxation time.
Equation (\ref{5}) is a conservation law for $\rho$.
(\ref{6}) is a rate equation for $v$, which is not
a conservation of momentum as in fluid flow equations.
The anticipation factor $g$ in (\ref{6})
compare to pressure in the momentum equation.
It describes drivers' car-following behavior.
The right hand side of (\ref{6}) is the relaxation term.
Let
\begin{eqnarray}
h(\rho,v)=\frac{v_e(\rho)-v}{\tau}.\label{hh}
\end{eqnarray}
When the state is in equilibrium, the system of
equations (\ref{5}) (\ref{6}) is reduced to the equilibrium equation
\begin{eqnarray}
\rho_t+(\rho v_e(\rho))_x=0\label{7}
\end{eqnarray}
with initial data
\begin{eqnarray}
\rho(x,0)=\rho_0(x)>0.\label{i3}
\end{eqnarray}
It is assumed that the equilibrium velocity $v_e(\rho)$
is a decreasing function of $\rho$, $v'_e(\rho)<0$.
It is also assumed that $v_e(0)=v_f$ and $v_e(\rho_j)=0$
where $v_f$ is the free flow
speed and $\rho_j$ is the jam concentration.
The equilibrium flux $q(\rho)=\rho v_e(\rho)$ is assumed to be a
concave function of $\rho$
\begin{eqnarray}
q''(\rho)=\rho v''_e(\rho)+2v'_e(\rho)<0.\label{conca}
\end{eqnarray}
The equilibrium characteristic speed is
\begin{eqnarray}
\lambda_*(\rho)=q'(\rho)=v_e(\rho)+\rho v'_e(\rho).\label{eqsp}
\end{eqnarray}
For the traffic flow model (\ref{5}) (\ref{6}),
the characteristic speeds are
\begin{eqnarray}
\lambda_1(\rho,v)=\rho v_e'(\rho)+v
<-\rho v_e'(\rho)+v=\lambda_2(\rho,v)\label{chars}
\end{eqnarray}
and the right eigenvectors of the Jacobian of the flux are
\[
r_i(\rho, v)=(1,(-1)^{i-1}v_e'(\rho))^T,\;\; i=1, \;2.
\]
The system is strictly hyperbolic provided $\rho>0$. Furthermore,
each characteristic field is genuinely nonlinear
\[
\nabla \lambda_i(\rho, v)\cdot r_i(\rho, v)=(-1)^{i-1} q''(\rho)
\neq 0, \;\; i=1,\; 2
\]
where the concavity of $q$ is assumed, see (\ref{conca}).
On the equilibrium curve $v=v_e(\rho)$, a marginal stability condition
\begin{eqnarray}
\lambda_1=\lambda_*<\lambda_2\label{8}
\end{eqnarray}
is satisfied.
Thus there is no
diffusion in the process of relaxation for the traffic flow model (\ref{5})
(\ref{6}).
(\ref{8}) is a direct consequence of the anisotropic feature
of traffic flows.
In Li \cite{Li}, using a generalized Glimm scheme,
we obtained global existence of solution of (\ref{5}) (\ref{6}) (\ref{i2})
for initial data of bounded total variation, of bounded oscillations
and of small distance to the equilibrium curve.
We also showed that a sequence of the solutions obtained for
the relaxed system converge to a solution of the
equilibrium equation (\ref{7}) as the relaxation parameter
goes to zero.
In the current paper, we study the continuous dependence of
the solution on its initial data in $L^1$ topology. The uniqueness of
solutions is a corollary of the continuous dependence of
the solution on its initial data.
We construct a functional for two solutions such that it is
equivalent to the $L^1$ distance between the two solutions
and it is time-decreasing. The construction makes use of
the $L^1$ contraction semigroup property for the scalar
conservation laws, Keyfitz \cite{BarK}, Kruzkov \cite{Kruz},
Lax \cite{Lax} %, Liu \cite{Lius}
and the exponential decay property of the source term.
We show an $L^1$-contractive property of the entropy solution operator
in the Riemann invariant coordinate.
For general systems of conservation laws, there is no such a property,
see \cite{Temple1}.
We show the $L^1$ stability of the equilibrium shock waves.
That is, the entropy solution with initial data as certain $L^1$-bounded
perturbations of an equilibrium shock wave exists globally and
tends to a shifted equilibrium shock wave in $L^1$ norm
as $t\to \infty$.
We then show that if the initial data $\rho_0$
is bounded and of compact support,
the entropy solution converges
in $L^1$ to an equilibrium $N$-wave as $t\to +\infty$.
Uniqueness issues do not seem to have been systematically studied
in conjunction with higher order models.
In general, the zero relaxation limit is highly
singular because of shock and initial layers.
In \cite{Nata}, Natalini obtained the uniqueness of the zero relaxation
for semilinear systems of equations with relaxation.
The uniqueness problem for the quasilinear case
remains open.
For the quasilinear system of equations (\ref{5}) (\ref{6}),
we obtain the $L^1$-convergence to and the uniqueness of
the zero relaxation limit.
We prove the uniqueness of the zero relaxation
by using the property that
the solution depends on its data continuously,
the fact that the signed distance $-v_e(\rho)+v$
of $(\rho,\; v)$ to the equilibrium curve is
one of the Riemann invariants and that it decays in $\tau$ exponentially.
The relaxation limit models
dynamic limit from the continuum nonequilibrium processes to
the equilibrium processes. Typical examples for the limit include
gas flow near thermal-equilibrium and phase transition with
small transition time.
There has been a large literature on the mathematical theory
of relaxation, see Chen, Levermore
and Liu \cite{CDL}, Liu \cite{Liu2},
Natalini \cite{Nata}.
The plan of the paper is the following:
In Section 2, we give the preliminaries including a brief
derivation of the traffic flow model.
In Section 3, we establish the $L^1$-contractivity property for solutions of
(\ref{5}) (\ref{6}).
Asymptotic behavior of solutions is studied in Section 4.
In Section 5, we obtain the $L^1$-convergence to and the uniqueness of
the zero relaxation limit.
In Section 6, we give the conclusions.
\section {Preliminaries}
Zhang's traffic flow model (\ref{5}) (\ref{6}) was derived based on
the physical assumption that
the time needed for a {\it following} vehicle to assume a
certain speed is determined by {\it leading} vehicles, see \cite{LiZ}
\cite{Zhang}.
For $\tau>0$ and $\Delta x>0$,
\[
\frac{dx}{dt}(t+\tau)=v_e(\rho(x+\Delta x,t)).
\]
To leading order
\[
v+\tau \frac{dv}{dt}=v_e(\rho(x,t))+\Delta x \rho_x v_e'(\rho(x,t)).
\]
That is
\begin{eqnarray}
\frac{dv}{dt}=\frac{v_e(\rho(x,t))-v}{\tau}+\frac{\Delta x }{\tau}
\rho_x v_e'(\rho(x,t)).\label{der}
\end{eqnarray}
Letting
\[
\frac{\Delta x}{\tau}=-(\lambda_*(\rho)-v_e(\rho))=-\rho v_e'(\rho),
\]
the relative wave propagating speed to the car speed
at the equilibrium, we obtain the anticipation factor which expresses
the effect of drivers reacting to conditions downstream.
The minus sign on the right hand side
comes from the fact that the behavior of the driver
is determined by leading vehicles.
We assume that
the equilibrium velocity $v_e(\rho)$ is a linear function
of $\rho$
\begin{eqnarray}
v_e(\rho)=-a \rho+b,\;\;a,\; b>0\label{ve}
\end{eqnarray}
as in \cite{Li}.
Under assumption (\ref{ve}),
\begin{eqnarray}
g(\rho)=\frac{a^2}{2}\rho^2\label{G1}
\end{eqnarray}
and
\begin{eqnarray}
q(\rho)=\rho v_e(\rho)= -a\rho^2+b \rho\label{q}
\end{eqnarray}
where the flux $q$ is a quadratic function which corresponds to
the flux of the classical PW(Payne-Whitham) model.
Therefore the case that the equilibrium velocity is linear
is an important nontrivial case in traffic flow.
The assumption has also been used by other authors, see, for example,
Lattanzio and Marcati \cite{LM}.
The right eigenvectors of the Jacobian of the flux of (\ref{5}) (\ref{6})
are
constant vectors
\[
r_i(\rho, v)=(1,(-1)^{i}a)^T,\;\; i=1, \;2.
\]
Thus both the rarefaction curves and shock curves are straight lines.
Furthermore, the shock wave curves coincide with the rarefaction wave
curves,
$S_i(u_0)= R_i(u_0)$, $\;i=1,2$.
Multiplying (\ref{5}) (\ref{6}) on the left
with the $j$th left eigenvector,
$l_j(\rho,v)=((-1)^{j-1}v_e'(\rho),1)^T,\;\; j=1, \;2 $, of the Jacobian
of the flux, we have that
\begin{eqnarray}
(-v_e(\rho)-v)_t+\lambda_1(-v_e(\rho)-v)_x=-h(\rho,v)\label{ch1}\\
(-v_e(\rho)+v)_t+\lambda_2(-v_e(\rho)+v)_x=h(\rho,v)\label{ch2}
\end{eqnarray}
where $h$ is defined in (\ref{hh}).
The Riemann invariants $r$ and $s$ are
\begin{eqnarray}
r(\rho, v)=-v_e(\rho)-v\label{ri1}\\
s(\rho, v)=-v_e(\rho)+v.\label{ri2}
\end{eqnarray}
From (\ref{ch2}) we see that
one of the Riemann invariants is
the signed distance $-v_e(\rho)+v$
of $(\rho,\; v)$ to the equilibrium curve.
Noting (\ref{hh}), (\ref{chars}) and (\ref{ve}), we have
\begin{eqnarray}
r_t-\left (\frac{1}{2}r^2+br\right)_x=\frac{s}{\tau}\label{RI11}\\
s_t+\left (\frac{1}{2}s^2+bs\right)_x=-\frac{s}{\tau}.\label{RI22}
\end{eqnarray}
The initial data is obtained from (\ref{i2})
\begin{eqnarray}
r(x,0)=r_0(x)\label{RI01}\\
s(x,0)=s_0(x).\label{RI02}
\end{eqnarray}
\section{The Cauchy problem}
For a scalar balance law
\begin{eqnarray}
u_t+f(x, t, u)_x=g(x, t, u)\label{scl}
\end{eqnarray}
with initial data
\begin{eqnarray}
u(x,0)=u_0(x),\label{scl0}
\end{eqnarray}
Kruzkov \cite{Kruz} defined generalized solutions of problem (\ref{scl})
(\ref{scl0}).
Let $\Pi_T=R\times [0,T]$.
Let $u_0(x)$ be a bounded measurable function such that $|u_0(x)|\leq M_0$
on $R$.
\paragraph{Definition} A bounded measurable function $u(x,t)$ is called
a generalized solution of problem (\ref{scl}) (\ref{scl0}) in $\Pi_T$ if:
i) for any constant $k$ and any smooth function $\phi(x,t)\geq 0$
which is finite in $\Pi_T$ (the support of $\phi$ is strictly in $\Pi_T$),
if the following inequality holds,
\[
\int\!\int_{\Pi_T} \{ |u(x,t)-k|\phi_t+\mathop{\rm sign}(u(x,t)-k)[f(x,t,
u(x,t))-f(x,t,k)]\phi_x-
\]
\begin{eqnarray}
-\mathop{\rm sign}(u(x,t)-k)[f_x(x,t, u(x,t))-g(x,t, u(x,t))]\}dxdt\geq 0;\label{ws}
\end{eqnarray}
ii) there exists a set $E$ of zero measure on $[0,T]$, such that for
$t\in[0,T]\backslash E$, the function $u(x,t)$ is defined almost
everywhere in $R$, and for any ball $K_r=\{|x|\leq r\}$
\[
\lim_{t\to 0}\int_{K_r}|u(x,t)-u_0(x)|dx=0.
\]
Inequality (\ref{ws}) is equivalent to condition $E$ in \cite{Olen},
if $(u_-, u_+)$ is a discontinuity of $u$ and $v$ is any number
between $u_-$ and $u_+$, then
\begin{eqnarray}
\frac{f(x, t, u_+)-f(x, t, u_-)}{u_+-u_-}\leq
\frac{f(x, t, v)-f(x, t, u_-)}{v-u_-}.
\label{Oen}
\end{eqnarray}
\paragraph{Remark} In the case that $f$ is strictly convex (or concave) in $u$
and $u_-\neq u_+$, the strict inequality in (\ref{Oen}) holds.
The following results on the existence and uniqueness of the generalized
solution of problem (\ref{scl}) (\ref{scl0}) are due to Kruzkov \cite{Kruz}.
Uniqueness follows from the following result on the stability
of the solutions relative to changes in the initial data in the norm of
$L^1$.
For any $R>0$ and $M>0$, we set
\[
N_M(R)=\max_{K_R\times [0,T]\times [-M,M]}|f_u(x,t,u)|
\]
and let $\kappa$ be the cone $\{(x,t): |x|\leq R-Nt, 0\leq t\leq
T_0=\min\{T, RN^{-1}\}\}$. Let $S_{\tau}$ designate the cross-section
of the cone $\kappa$ by the plane $t=\tau$, $\tau\in[0,T_0]$.
\begin{theorem}\label{Kruz1} (Kruzkov)
Assume that: i) $f(x,t,u)$ and $g(x,t,u)$ are continuously differentiable
in the region $\{(x,t)\in\Pi_T, -\infty__0$ %\in [0,T_0]$
\begin{eqnarray}
\int_{S_t}|s_1(x,t)-s_2(x,t)|dx\leq e^{-\frac{t}{\tau}}
\int_{S_0}|s_{10}(x)-s_{20}(x)|dx.\label{uniq1}
\end{eqnarray}
\end{theorem}
\paragraph{Proof}
Applying Theorem \ref{Kruz1} to two solutions, $s_1$ and $s_2$, of
equation (\ref{RI22}) and noting that $\gamma=-\frac{1}{\tau}<0$, we
obtain (\ref{uniq1}).
\hfill$\diamondsuit$\smallskip
We obtain a global bound for the generalized solutions of (\ref{RI22})
for bounded measurable data (\ref{RI02}).
\begin{theorem}\label{bound}
Generalized solutions to the initial value problem (\ref{RI22}) (\ref{RI02})
are bounded almost everywhere
and the bounds depend only on their initial data. % and $v_*(\rho,x)$.
\end{theorem}
\paragraph{Proof}
Let $s$ be a generalized solution of equation (\ref{RI22}).
Applying Theorem \ref{seq} to two solutions, $s$ and $0$, of
equation (\ref{RI22}), we
conclude that the generalized solutions are bounded almost everywhere
or all $t>0$.
\hfill$\diamondsuit$\smallskip
It can be checked that all conditions in Theorem \ref{Kruz3}
are satisfied by equation (\ref{RI22}). Therefore we have
the following global existence result.
\begin{theorem}\label{seq1}
A unique generalized solution of problem (\ref{RI22}) (\ref{RI02})
exists globally.
\end{theorem}
Now we turn to solve the initial value problem (\ref{RI11}) with
bounded measurable data (\ref{RI01}).
\begin{theorem}\label{req}
If $r_1(x,t)$ and $r_2(x,t)$ are generalized solutions of problem
(\ref{RI11}) (\ref{RI01}) with bounded measurable
initial data $r_{10}(x)$ and $r_{20}(x)$ such that $r_{10}-r_{20}\in L^1$.
Then for almost all $t>0$ %\in [0,T_0]$
\begin{eqnarray}
\int_{S_t}|r_1(x,t)-r_2(x,t)|dx
&\leq& \int_{S_0}|r_{10}(x)-r_{20}(x)|dx+ \label{uniq2}\\
&&+(1- e^{-\frac{t}{\tau}})
\int_{S_0}|s_{10}(x)-s_{20}(x)|dx. \nonumber
\end{eqnarray}
\end{theorem}
\paragraph{Proof}
Applying the proof of Theorem \ref{Kruz1} in \cite{Kruz} to problem
(\ref{RI11}) (\ref{RI01}) and using (\ref{uniq1}), we obtain (\ref{uniq2}).
\hfill$\diamondsuit$\smallskip
Similarly, we obtain a global bound for the generalized solutions of
(\ref{RI11}).
\begin{theorem}\label{bound1}
Generalized solutions to the initial value problem (\ref{RI11}) (\ref{RI01})
are bounded almost everywhere
and the bounds depend only on their initial data.
\end{theorem}
\paragraph{Proof}
Let $\rho_e$ be the bounded solution to
the equilibrium equation (\ref{7}) with initial data (\ref{i3}),
$r_e=-2v_e(\rho_e)$ and $s_e=0$.
Then $r_e$ is a bounded solution to
(\ref{RI11}) with initial data $-2v_e(\rho_e(x,0))$
and $s_e$ is a solution to (\ref{RI22}) with initial data zero.
Applying Theorem \ref{req} to two solutions $r$ and $r_e$ of (\ref{RI11}),
we obtain the global boundedness of the generalized solution $r$.
\hfill$\diamondsuit$\smallskip
Finally, we have the following.
\begin{theorem}\label{req1}
A unique generalized solution of problem (\ref{RI11}) (\ref{RI01})
exists globally.
\end{theorem}
We show an $L^1$-contractive property of the generalized solutions
of (\ref{RI11}) (\ref{RI22}) in terms of the Riemann invariants.
\begin{theorem}\label{contr}
If $(r_1(x,t), s_1(x,t))$ and $(r_2(x,t), s_2(x,t))$
are generalized
solutions of (\ref{RI11}) (\ref{RI22}) for all $x$ and $t>0$,
with initial data
$(r_{10}(x), s_{10}(x))$, $(r_{20}(x), s_{20}(x))$
which are bounded measurable
and $r_{10}-r_{20}, s_{10}-s_{20} \in L^1$,
then
\begin{eqnarray}
\lefteqn{ \|r_1(\cdot,t)-r_2(\cdot,t)\|_{L^1}+\|s_1(\cdot,t)
-s_2(\cdot,t)\|_{L^1} } \nonumber\\
&\leq & \|r_{10}(\cdot)- r_{20}(\cdot)\|_{L^1}+\|s_{10}(\cdot)-s_{20}(\cdot)
\|_{L^1}.
\label{contr1}
\end{eqnarray}
\end{theorem}
\paragraph{Proof}
Combining the results of Theorem \ref{seq} and Theorem \ref{req},
we arrive at our conclusion.
\hfill$\diamondsuit$\smallskip
\paragraph{Remark} It is interesting to note that there is no contractive property
for $r$, see (\ref{uniq2}). However, there is the contractive property for
$r$ and $s$, see (\ref{contr1}). This property allows us to
investigate the large-time behavior of solutions in next section.
From (\ref{ri1}) (\ref{ri2}), it is evident that
$\|r_1(\cdot,t)-r_2(\cdot,t)\|_{L^1}+
\|s_1(\cdot,t)-s_2(\cdot,t)\|_{L^1}$ is equivalent to
the $L^1$ distance of the two solutions.
Thus the $L^1$ well-posedness theory for the Cauchy problem
(\ref{5}) (\ref{6}) (\ref{i2}) is established.
\begin{theorem}\label{distance44}
If $(\rho_1(x,t), v_1(x,t))$ and $(\rho_2(x,t), v_2(x,t))$
are generalized solutions of (\ref{5}) (\ref{6}) for all $x$ and $t>0$,
with bounded measurable initial data
$(\rho_{10}(x), v_{10}(x))$, $(\rho_{20}(x), v_{20}(x))$
such that $\rho_{10}-\rho_{20}, v_{10}-v_{20} \in L^1$,
then
\begin{eqnarray}
\lefteqn{ \|\rho_1(\cdot,t)-\rho_2(\cdot,t)\|_{L^1}+\|v_1(\cdot,t)
-v_2(\cdot,t)\|_{L^1} }\nonumber\\
&\leq &
C(\|\rho_{10}(\cdot)-\rho_{20}(\cdot)\|_{L^1}+\|v_{10}(\cdot)-v_{20}(\cdot)
\|_{L^1})
\label{distance4}
\end{eqnarray}
where $C$ is a constant independent of $t$ and the initial data.
\end{theorem}
\section{Asymptotic Behavior}
We study the large-time behavior of the entropy
solutions of (\ref{5}) (\ref{6}).
We show that the entropy solutions with initial data as certain $L^1$
bounded
perturbations of an equilibrium shock wave exist and
tend to a shifted equilibrium shock wave in $L^1$ norm
as $t\to \infty$.
Recall that a steady-state solution is
either a constant state on the equilibrium curve, i.e.,
$(\rho,v)=(\rho_0,v_e(\rho_0))$
or an equilibrium shock wave
\begin{eqnarray}
(\rho_{sh},v_{sh})(x)=\left\{
\begin{array}{ll}
(\rho_-,v_e(\rho_-))\ \;\; &x\leq x_0\\
(\rho_+,v_e(\rho_+))&x> x_0
\end{array}\right.\label{stsh}
\end{eqnarray}
satisfying the entropy condition $\rho_-\leq \rho_+$.
Denote the Riemann invariants of the equilibrium shock wave
by $R(x)$ and $S(x)$, see (\ref{ri1}) (\ref{ri2}).
We show that the equilibrium shock waves are nonlinearly stable in $L^1$
norm.
The main tools used in the proof
are the $L^1$ contractivity, a result of Kruzkov \cite{Kruz} and
the exponentially decay property of the source term, see Theorem \ref{seq}.
Without loss of generality, we set
\begin{eqnarray}
(\rho_0(-\infty), v_0(-\infty))=(\rho_-, v_e(\rho_-)),\;
(\rho_0(+\infty), v_0(+\infty))=(\rho_+, v_e(\rho_+)). \label{RP1}
\end{eqnarray}
\begin{theorem}\label{aym}
Let the initial data (\ref{i2}) be a bounded perturbation of
an equilibrium shock wave, satisfy
\begin{eqnarray}
(\rho_{0}-\rho_{sh}, v_0-v_{sh})\in (L^\infty)^2\cap (L^1)^2,\label{l1c}
\end{eqnarray}
be of small distance to the equilibrium curve and satisfy further that
\begin{eqnarray}
R(-\infty)\leq r(x,0)\leq R(+\infty).\label{l1c2}
\end{eqnarray}
Then the global bounded entropy solution of
(\ref{5}) (\ref{6}) (\ref{i2}) exists and
tends to a shifted equilibrium shock wave
in $L^1$ norm as $t\to \infty$,
\begin{eqnarray}
\lim_{t\to +\infty}(\|\rho(\cdot,t)-\rho_{sh}(\cdot+k)\|_{L^1}
+\|v(\cdot,t)-v_{sh}(\cdot+k)\|_{L^1})=0\label{concl}
\end{eqnarray}
where the shift $k$ is given by
\begin{eqnarray}
k=\frac{1}{\rho_{sh}(+\infty)-\rho_{sh}(-\infty)}\int_R
(\rho(x,0)-\rho_{sh}(x))
dx.\label{mass}
\end{eqnarray}
\end{theorem}
We state a result of Kruzkov \cite{Kruz} before we prove Theorem \ref{aym}.
Consider the initial value problem
\begin{eqnarray}
u_t+(f(u))_x&=&g\label{scls}\\
u(x,0)&=&u_0.\label{iscls}
\end{eqnarray}
Let $\Sigma_T=R\times (0,T)$, $\displaystyle{[a]_+=\frac{1}{2}(|a|+a)}$ and
$\displaystyle{H=H(a)= \frac{1}{2}(\mathop{\rm sgn} a+1),}$
is the Heavyside function.
\begin{theorem}\label{Kruz2}
Let $u$($v$) be an entropy subsolution(supersolution)
of (\ref{scls}) (\ref{iscls}) in $\Sigma_T$ for the right-hand side $g$($h$)
and the initial data $u_0$($v_0$).
Fix $a$, $b$ such that $u, v\in [a,b]$ in $\Sigma_T$.
Then for each interval $(\alpha,\beta)$, we have
\begin{eqnarray}
\lefteqn{ \int_{\alpha+tK}^{\beta-tK}[u(x,t)-v(x,t)]_+dx }\nonumber\\
&\leq& \int_{\alpha}^{\beta}[u_0(x)-v_0(x)]_+dx+ \label{kruz}\\
&&+\int_0^t ds \int_{\alpha+tK}^{\beta-tK}H(u(x,s)-v(x,s))(g(x,s)-h(x,s))dx
\nonumber
\end{eqnarray}
where $\displaystyle{K\geq L=\|f'(u)\|_{L^{\infty}(a,b)},
00$
and $\frac{\partial g_2}{\partial r}=0$.
The conclusion is proved.
\hfill$\diamondsuit$\smallskip
Therefore, solutions of (\ref{RI11}) (\ref{RI22}) satisfy a comparison
principle, see \cite{Nata}.
\begin{theorem}
Let $ U^1$ and $ U^2$ be two weak solutions of (\ref{RI11}) (\ref{RI22})
in $\mathbb{R}\times (0,T)$ with initial data $U^1_0$ and $U_0^2$ respectively.
If $U^1_0\leq U_0^2$ for almost every $x\in \mathbb{R}$, then
$U^1\leq U^2$ for almost every $(x,t)\in \mathbb{R}\times (0,T)$.
\end{theorem}
Now we prove Theorem \ref{aym}.
\paragraph{Proof}
Boundedness of the solution follows directly from the quasimonotone
property of the source terms of (\ref{RI11}) (\ref{RI22})
and the comparison principle.
Step 1. We first prove the conclusion for initial data (\ref{RI01})
(\ref{RI02}) satisfying
some additional ordering properties besides (\ref{l1c}) (\ref{l1c2}).
Let
\[
R(x)=r(\rho_{sh},v_{sh})=-2v_e(\rho_{sh})
\]
and
\[
S(x)=s(\rho_{sh},v_{sh})=0
\]
be stationary solutions of (\ref{RI11}) and (\ref{RI22}) respectively.
Let $r(x,0)$ satisfy
\begin{eqnarray}
R(x+\gamma)\leq r(x,0)\leq R(x+\beta), \mbox{ for all}\;\; x\label{mono}
\end{eqnarray}
for some $\gamma$ and $\beta$.
Let
\[
r(x,0)=R(x)+\psi_0(x)
\]
and
\[
r(x,t)=R(x)+\psi(x,t).
\]
Applying Theorem \ref{Kruz2} to solutions of (\ref{RI11})
and noting (\ref{mono}), we have that
\[
R(x+\gamma)\leq r(x,t)\leq R(x+\beta)
\]
for all $x$ and for all $t>0$ %with $T$ large
or
\begin{eqnarray}
R(x+\gamma)-R(x)\leq \psi(x,t)\leq R(x+\beta)-R(x)
\label{mono1}
\end{eqnarray}
for all $x$ and for all $t>0$.
(\ref{mono1}) follows from (\ref{kruz}), that the initial data
is of small distance (relative to the equilibrium shock wave strength)
to the equilibrium curve and that (\ref{RI11})
has a source term that decays exponentially in $t$, see (\ref{uniq1}).
%This is due to the fact that (\ref{RI11})
%has a source term that decays exponentially in $t$, see (\ref{distance1})
%and (\ref{kruz}).
%We have thus proved an asymptotic ordering property for the solutions.
Therefore $\{\psi(x,t)\}_{t>0}$ is uniformly bounded by functions in $L^1$.
We claim that $\{\psi(x,t)\}_{t>0}$ is
$L^1$-equicontinuous. In fact,
\begin{eqnarray*}
\lefteqn{ \|\psi(\cdot+h,t)-\psi(\cdot,t)\|_{L^1} }\\
&\leq&\|r(\cdot+h,t)-r(\cdot,t)\|_{L^1}+\|R(\cdot+h)-R(\cdot)\|_{L^1}\\
&\leq&\|r(\cdot+h,0)-r(\cdot,0)\|_{L^1}+(1-e^{-\frac{t}{\tau}})
\|s(\cdot+h,0)-s(\cdot,0)\|_{L^1}+\\
&&+\|R(\cdot+h)-R(\cdot)\|_{L^1}\to 0
\end{eqnarray*}
as $h\to 0$, uniformly with respect to $t>0$, where we have used
the continuous dependence on data property (\ref{uniq2})
and the condition on the initial data (\ref{l1c}).
Hence $\{\psi(x,t)\}_{t>0}$ is relatively compact in $L^1$.
Let $B_s$ be the set of accumulation points of $\{\psi(x,t)\}_{t>s}$ for
$s>0$,
then $B_s$ is not empty by compactness. Hence
\[
A=R(x)+\cap_{s\geq 0}B_s\neq \emptyset
\]
represents the set of all the possible $L^1$-limit of $r(x,t)$
as $t\to +\infty$.
It is enough to prove that $A=\{R(x+k)\}$ for some $k$.
Let $a^0(x)\in A$. Then there exist $t_n>0$ such that $t_n\to +\infty$
as $n\to +\infty$ and
\begin{eqnarray}
\lim_{n\to +\infty}\|r(\cdot, t_n)-a^0(\cdot)\|_{L^1}=0.\label{lim0}
\end{eqnarray}
>From Theorem \ref{req}, we know that
$\|r_1(\cdot,t)-r_2(\cdot,t)\|_{L^1}
+e^{-\frac{t}{\tau}}\|s_{10}(\cdot)-
s_{20}(\cdot)\|_{L^1}$ decreases in $t$ for any two solutions and
hence it admits limit as $t\to +\infty$.
Therefore for any $h$,
\begin{eqnarray}
\lim_{t\rightarrow +\infty} \|r(\cdot,t)-R(\cdot+h)\|_{L^1}=c_h\label{alim}
\end{eqnarray}
for some $c_h\geq 0$.
Letting $t_n\to +\infty$ in the above limit, we have
\[
\|a^0(\cdot)-R(\cdot+h)\|_{L^1}=c_h.
\]
Let $a(x,t)$ be the solution of (\ref{RI11}) with initial data $a^0(x)$.
Then $a(x,t)\in A$ ($A$ is invariant under the flow defined by
(\ref{RI11})).
Therefore for the same reason
\[
\|a(\cdot,t)-R(\cdot+h)\|_{L^1}=c_h
\]
for any $h$ and any $t>0$.
Applying the contractive property
(\ref{contr1}) to two solutions $(a(x,t), 0)$ and $(R(x+h), 0)$
of (\ref{RI11}) (\ref{RI22}), we have that
\[
\begin{array}{rl}
0&=\|a(\cdot,t)-R(\cdot+h)\|_{L^1}-\|a^0(\cdot)-R(\cdot+h)\|_{L^1}\\
&\leq 0
\end{array}
\]
for any $h$ and any $t>0$. From the proof of Theorem \ref{seq}
and Theorem \ref{req} we know that the above equality
holds if and only if $a(x,t)-R(x+h)$ has no shock
at the point of sign change for any $h$ and any $t>0$, see the remark
follows (\ref{Oen}).
This shows that there exists some $k$, satisfying (\ref{mass}) due to
conservation law (\ref{5}),
such that for any $t>0$
\[
a(x,t)=R(x+k).
\]
Thus there exists $t_n\to +\infty$ as $n\to +\infty$
such that
\[
\lim_{n\to +\infty}\|r(\cdot, t_n)-R(\cdot+k)\|_{L^1}=0.
\]
By (\ref{alim}),
$\|r(\cdot, t)-R(\cdot+k)\|_{L^1}$ admits a limit as $t\to +\infty$,
we conclude that $r(x, t)-R(x+k)$ converges to $0$ in $L^1$
as $t\to +\infty$. By (\ref{uniq1}), $\|s(\cdot, t)-S(\cdot+k)\|_{L^1}$
decays to zero exponentially in $t$.
Thus $\|r(\cdot,t)-R(\cdot+k)\|_{L^1}+
\|s(\cdot,t)-S(\cdot+k)\|_{L^1}$ converges to zero.
On the other hand, $\|r(\cdot,t)-R(\cdot+k)\|_{L^1}+
\|s(\cdot,t)-S(\cdot+k)\|_{L^1}$ is equivalent to
the $L^1$ distance of the two solutions, we arrive at the conclusion
(\ref{concl}).
Step 2. For general initial data satisfying (\ref{l1c}) (\ref{l1c2}), we have
\[
R(-\infty)\leq r(x,0)=R(x)+\psi_0(x)\leq R(+\infty),
\]
where $\psi_0\in L^1(R)$.
Let $\psi_0^n\in L^1(R)$ be a sequence of functions satisfying
the ordering properties (\ref{mono}) defined in Step 1
and
\[
\lim_{n\to \infty}\|\psi_0-\psi_0^n\|_{L^1}=0.
\]
Let $r^n(x,t)$ and $s^n(x,t)$
be the solutions of the initial value problem (\ref{RI11}) (\ref{RI22})
with $r^n(x,0)=R(x)+\psi_0^n(x)$ and $s^n(x,0)=s(x,0)$ respectively.
Then, by Step 1, there exist $k_n$ such that
\[
\lim_{t\to \infty}\|r^n(\cdot, t)-R(\cdot+k_n)\|_{L^1}=0
\]
for each $n$.
By the contractive property (\ref{contr1}) and
that $r^n(\cdot, 0)$ and $s^n(\cdot, 0)$
are Cauchy sequences in $L^1$, we deduce that $r^n(\cdot, t)$ and $s^n(\cdot, t)$
are Cauchy sequences for any $t>0$. Therefore, by letting
$t\to \infty$, we obtain that $R(\cdot+k_n)$ is a Cauchy sequence
too.
So
\[
\int_R |R(x+k_n)-R(x+k_m)|dx=(R(+\infty)-R(-\infty))|k_n-k_m|.
\]
Thus, for any $\epsilon>0$, there is an $N$, such that if $m,n>N$, then
\[
|k_n-k_m|<\epsilon.
\]
Therefore
\[
\lim_{n\to \infty}k_n=k
\]
\[
\lim_{n\to \infty}\|R(\cdot+k_n)-R(\cdot+k)\|_{L^1}=0
\]
for some $k$.
Finally,
\[
\|r(\cdot, t)-R(\cdot+k)\|_{L^1}\leq \|r(\cdot, t)-r^n(\cdot, t)\|_{L^1}+
\]
\[
+\|r^n(\cdot, t)-R(\cdot+k_n)\|_{L^1}+\|R(\cdot+k_n)-R(\cdot+k)\|_{L^1}.
\]
Therefore,
we conclude that $r(x, t)$ converges to $R(x+k)$ in $L^1$
as $t\to +\infty$. By (\ref{uniq1}), $\|s(\cdot, t)-S(\cdot+k)\|_{L^1}$
decays to zero exponentially in $t$.
Thus $\|r(\cdot,t)-R(\cdot+k)\|_{L^1}+
\|s(\cdot,t)-S(\cdot+k)\|_{L^1}$ converges to zero.
Since $\|r(\cdot,t)-R(\cdot+k)\|_{L^1}+
\|s(\cdot,t)-S(\cdot+k)\|_{L^1}$ is equivalent to
the $L^1$ distance of the two solutions, we arrive at the conclusion that
the entropy solution of
(\ref{5}) (\ref{6}) (\ref{i2}) exists globally and
tends to a shifted equilibrium shock wave
in $L^1$ norm as $t\to \infty$, where the shift $k$ satisfies
(\ref{mass}) due to conservation law (\ref{5}).
\hfill$\diamondsuit$\smallskip
Now we consider bounded compact support initial data $\rho_0$ in
(\ref{i2}).
We show that the entropy solution converges
in $L^1$ to an equilibrium $N$-wave as $t\to +\infty$.
An $N$-wave of a scalar conservation law (\ref{scl}) is
\begin{eqnarray}
N(x,t)=\left\{
\begin{array}{ll}
\frac{1}{k}\left(\frac{x}{t}-c\right)\;\; &-(pkt)^{\frac{1}{2}}0$. %The constant $C$ depends
$v\to v_e(\rho)$ in $L^1$ norm as $t\to +\infty$.
\end{theorem}
\paragraph{Proof}
Consider two entropy solutions, $(\rho, v)$
and $(\rho_e, v_e(\rho_e))$, of (\ref{5}) (\ref{6}),
where $\rho_e$ is the unique entropy solution of the equilibrium
equation (\ref{7}) with initial data
$\rho_0$. Applying Theorem \ref{seq} to these two solutions, we have
\begin{eqnarray}
\|s_1(\cdot,t)-s_2(\cdot,t)\|_{L^1}\leq e^{-\frac{t}{\tau}}\|s_{10}(\cdot)-
s_{20}(\cdot)\|_{L^1}\to 0 \label{dist111}
\end{eqnarray}
as $t\to +\infty$,
where $\displaystyle {s_1=-v_e(\rho)+v}$
and $\displaystyle {s_2=0}$.
We claim that $\{\rho(x, t)-\rho_e(x, t)\}_{t>0}$ is
$L^1$-equicontinuous. In fact,
\begin{eqnarray*}
\lefteqn{ \|\rho(\cdot+h,t)-\rho_e(\cdot+h,t)-\rho(\cdot,t)
+\rho_e(\cdot,t)\|_{L^1} }\\
&\leq&
\|\rho(\cdot+h,t)-\rho(\cdot,t)\|_{L^1}+\|\rho_e(\cdot+h,t)-\rho_e(\cdot,t)
\|_{L^1}\\
&\leq &C(\|r(\cdot+h,t)-r(\cdot,t)\|_{L^1}+
\|s(\cdot+h,t)-s(\cdot,t)\|_{L^1})+\\
&&+\|\rho_e(\cdot+h,t)-\rho_e(\cdot,t)\|_{L^1}\\
&\leq &C(\|r(\cdot+h,0)-r(\cdot,0)\|_{L^1}+
\|s(\cdot+h,0)-s(\cdot,0)\|_{L^1})+\\
&&+\|\rho_0(\cdot+h)-\rho_0(\cdot)\|_{L^1}\to 0
\end{eqnarray*}
uniformly with respect to $t>0$ as $h\to 0$, where we have used
the continuous dependence on data property (\ref{contr1}).
Hence $\{\rho(x, t)-\rho_e(x, t)\}_{t>0}$ is relatively compact in $L^1$.
Let $A$ be the set of accumulation points of $\{\rho(x, t)-\rho_e(x, t)\}_{t>0}$,
then $A\subset L^\infty\cap L^1$ is not empty by compactness.
Let $\phi(x)\in A$, then $\phi(x)$ is of compact support
and there exists a sequence $t_n$ such that
$t_n\to +\infty$ as $n \to +\infty$ and
\[
\|\rho(\cdot, t_n)-\rho_e(\cdot, t_n)-\phi(\cdot)\|_{L^1}\to 0.
\]
Letting $t_n\to +\infty$ in (\ref{5}) and
noting (\ref{dist111}) and the uniform boundedness of $(\rho, v)$,
we have that $\rho(x, t_n)$ solves (\ref{7}) asymptotically.
%The error is of order $e^{-\frac{t_n}{\tau}}$ for $t_n$ large
%as indicated in (\ref{dist111}).
Noticing that $\rho_e$ is the unique entropy solution of (\ref{7}) with data $\rho_0$
and that $\phi(x)$ is of compact support,
we deduce that $\phi(x)=0$ a.e..
That is $A=\{0\}$. Since every convergent sequence of
$\rho(x, t)-\rho_e(x, t)$ converges to a same limit $0$,
therefore, we have
\[
\|\rho(\cdot, t)-\rho_e(\cdot, t)\|_{L^1}\to 0
\quad
\mbox{as }t\to +\infty\,.
\]
On the other hand, $\rho_e(x,t)$ decays in $L^1$ to the $N$-wave $N(x,t)$
determined by the initial data at a rate $t^{-\frac{1}{2}}$
as $t\to +\infty$.
Furthermore, $\rho$ decays to the $N$-wave also at a rate $t^{-\frac{1}{2}}$
as $t\to +\infty$.
\section{Unique Zero Relaxation Limit}
Uniqueness issues do not seem to have been systematically studied
in conjunction with higher order models.
In general, the zero relaxation limit is highly
singular because of shock and initial layers.
In \cite{Nata}, Natalini obtained the uniqueness of the zero relaxation
for semilinear systems of equations with relaxation.
The uniqueness problem for the quasilinear case
remains open.
For the quasilinear system of equations (\ref{5}) (\ref{6}),
we show that the entropy
solutions of (\ref{5}) (\ref{6}) (\ref{i2})
%obtained for the relaxed system
converge in $L^1$ norm to the unique
entropy solution of the
equilibrium equation (\ref{7}) (\ref{i3})
as the relaxation parameter $\tau$ goes to zero.
The limit models
dynamic limit from the continuum nonequilibrium processes to
the equilibrium processes.
%In general, the zero relaxation limit is highly
%singular because of shock and initial layers, see
%Chen, Levermore and Liu \cite{CDL}.
We proved the uniqueness of the zero relaxation limit
by using the property that the solution depends on its data
continuously,
the fact that the signed distance $-v_e(\rho)+v$
of $(\rho,\; v)$ to the equilibrium curve is
one of the Riemann invariants and that it decays in $\tau$ exponentially.
We denote the solutions to (\ref{5}) (\ref{6}) (\ref{i2})
as $(\rho^\tau, v^\tau)$ for each $\tau>0$ and
$\rho$ the unique entropy solution of the equilibrium
equation (\ref{7}) (\ref{i3}).
\begin{theorem}\label{un1}
Let $(\rho^\tau, v^\tau)$ be the global bounded entropy
solution of (\ref{5}) (\ref{6}) (\ref{i2})
with $(\rho_0, v_0)\in (L^\infty)^2$
and $v_0-v_e(\rho_0)\in L^1$.
Then $(\rho^\tau, v^\tau)$
converges in $(L^1)^2$ to
$(\rho,v_e(\rho))$ as $\tau\to 0$ for any $t>0$. Moreover,
$\rho$ %has bounded total variation
is the unique entropy solution of the equilibrium
equation (\ref{7}) (\ref{i3}).
\end{theorem}
\paragraph{Proof}
Let $(\rho^\tau, v^\tau)$ be the unique entropy solution of
(\ref{5}) (\ref{6}) (\ref{i2}).
Let $\rho$ be the unique entropy
solution of the equilibrium
equation (\ref{7}) (\ref{i3}).
Applying (\ref{uniq1}) to the two solutions $(\rho^\tau, v^\tau)$
and $(\rho, v_e(\rho))$, we have that
\begin{eqnarray}
\|s_1^\tau(\cdot,t)-s_2(\cdot,t)\|_{L^1}\leq e^{-\frac{t}{\tau}}\|s_{10}(\cdot)-
s_{20}(\cdot)\|_{L^1}\to 0\label{uniq3}
\end{eqnarray}
as $\tau\to 0$ for $t>0$,
where $\displaystyle {s_1^\tau=-v_e(\rho^\tau)+v^\tau}$ and $\displaystyle
{s_2=0}$.
Therefore
\begin{eqnarray}
\|-v_e(\rho^\tau)+v^\tau\|_{L^1}\to 0\label{vcon}
\end{eqnarray}
as $\tau\to 0$ for $t>0$.
%Since $s^\tau_1$ is an entropy solution to a single conservation law
%(\ref{RI22}), hence
%\[
%\lim_{t\to 0}\lim_{\tau\to 0}\|-v_e(\rho^\tau)+v^\tau\|_{L^1}=
%\lim_{t\to 0}\lim_{\tau\to 0}\|s^\tau_1(\cdot,t)\|_{L^1}=0.
%\]
Applying Theorem \ref{contr} to two solutions, $(r_1^\tau, s_1^\tau)$
and $(r_2, s_2)$, of (\ref{RI11}) (\ref{RI22}),
where $r_1^\tau(\rho^\tau, v^\tau)=-v_e(\rho^\tau)-v^\tau$ and
$r_2(\rho, v_e(\rho))=-2v_e(\rho)$,
we have
that the $(L^1)^2$ distance between these two solutions
is uniformly bounded with respect to the
relaxation parameter $\tau$.
We claim that $\{r_1^\tau(x,t)-r_2(x,t)\}_{\tau>0}$ is
$L^1$-equicontinuous in $x$ and
locally $L^1$ Lipschitz continuous in $t$.
The $L^1$-equicontinuity in $x$ is obtained
by using the continuous dependence on data property (\ref{contr1}),
see the proof of Theorem \ref{Asy}.
The locally
$L^1$ Lipschitz continuous in $t$ for $t>t_\tau=\tau \ln \frac{1}{\tau}$
is a direct consequence of finite speed of propagation of
the elementary waves
and the exponential decay in $\tau$ of the source terms of
(\ref{RI11}) (\ref{RI22}),
see \cite{Kruz} \cite{Nata}.
Therefore, there is a sequence $\tau_n$ such that $\tau_n \to 0$
as $n\to +\infty$ and that $r_1^{\tau_n}(x,t)-r_2(x,t)$
converges to a function %denoted as $\phi(x,t)$
for each $t>0$.
Combining with (\ref{uniq3}), we have
that as $n\to +\infty$, $\rho^{\tau_n}(x, t)-\rho(x, t)$
converges to a function denoted as $\phi(x,t)$ for $t>0$.
It can be checked that
$\phi(x,t)\in L^\infty\cap L^1$ for all $t>0$.
Noticing that $\rho^{\tau_n}$ and $\rho$ have the same initial data
(\ref{i2}) (\ref{i3}), we have that $\phi(x,0)=0$.
Letting $\tau_n\to 0$ in (\ref{7}) and
noting the uniform boundedness of $(\rho^{\tau_n}, v^{\tau_n})$ and
(\ref{uniq3}),
we derive that the limit $\phi(x,t)=0$ a.e..
Therefore
\[
\|\rho^{\tau_n}(\cdot, t)-\rho(\cdot, t)\|_{L^1}\to 0
\]
as $n\to +\infty$.
Since every convergent sequence of $\rho^\tau(x,t)-\rho(x,t)$ converges to
a same limit $0$,
we conclude that $\rho^\tau-\rho$ converges to $0$ in $L^1$
as $\tau\to 0$. This and (\ref{vcon}) prove the theorem.
\section{Conclusions}
For a nonequilibrium continuum traffic flow model, which was
derived based on the assumption that
drivers respond with a delay to changes of traffic conditions
in front of them,
we established the $L^1$ well-posedness theory
for the Cauchy problem.
We obtained the continuous dependence
of the solution on its initial data in $L^1$ topology.
We constructed a functional for two solutions which is equivalent to
the $L^1$ distance between the solutions.
We proved that the functional
decreases in time which yields the $L^1$ well-posedness of the Cauchy
problem.
We also showed that the equilibrium shock waves are nonlinearly stable
in $L^1$ norm.
That is, the entropy solution with initial data as certain $L^1$ bounded
perturbations of an equilibrium shock wave exists globally and
tends to a shifted equilibrium shock wave in $L^1$ norm
as $t\to \infty$.
We then showed that if the initial data
$\rho_0$ is bounded and of compact support,
the entropy solution converges
in $L^1$ to an equilibrium $N$-wave as $t\to +\infty$.
We finally showed that the solutions for the relaxed
system converge in the $L^1$ norm
to the unique entropy solution of the equilibrium equation as the relaxation
time goes to zero.
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\end{thebibliography}
\noindent{\sc Tong Li }\\
Department of Mathematics, University of Iowa\\
Iowa City, IA 52242, USA \\
Tele: (319)335-3342 Fax: (319)335-0627\\
e-mail: tli@math.uiowa.edu
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