We establish the 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 topology. We construct a functional for two solutions which is equivalent to the distance between the solutions. We prove that the functional decreases in time which yields the well-posedness of the Cauchy problem. We thus obtain the -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 norm. That is, the entropy solution with initial data as certain -bounded perturbations of an equilibrium shock wave exists globally and tends to a shifted equilibrium shock wave in norm as . We also show that if the initial data is bounded and of compact support, the entropy solution converges in to an equilibrium -wave as .
Submitted June 2, 2000. Published February 20, 2001.
Math Subject Classifications: 5L65, 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.
Show me the PDF file (277K), TEX file, and other files for this article.