L1 stability of conservation laws for a traffic flow model

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 ->infinity. 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 ->+infinity.