L^1 stability of conservation laws for a traffic flow model

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.

Keywords

• Relaxation
• shock
• rarefaction
• $L^1$-contraction
• traffic flows
• anisotropic
• equilibrium
• marginally stable
• zero relaxation limit
• large-time behavior
• $L^1$-stability.