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.