Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 14, pp. 1-18.
Title: $L^1$ stability of conservation laws for a traffic flow model
Author: Tong Li (Univ. of Iowa, Iowa City, USA)
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
$t\to +\infty$.
Submitted June 2, 2000. Published February 20, 2001.
Math Subject Classifications: 35L65, 35B40, 35B50, 76L05, 76J10.
Key Words: Relaxation; shock; rarefaction; $L^1$-contraction;
traffic flows; anisotropic; equilibrium; marginally stable;
zero relaxation limit; large-time behavior; $L^1$-stability.