Large-Time Behavior of Entropy Solutions of Conservation Laws

Abstract

We are concerned with the large-time behavior of discontinuous entropy solutions for hyperbolic systems of conservation laws. We present two analytical approaches and explore their applications to the asymptotic problems for discontinuous entropy solutions. These approaches allow the solutions of arbitrarily large oscillation without a priori assumption on the ways from which the solutions come. The relation between the large-time behavior of entropy solutions and the uniqueness of Riemann solutions leads to an extensive study of the uniqueness problem. We use a direct method to show the large-time behavior of largeL∞solutions for a class ofm×msystems including a model in multicomponent chromatography; we employ the uniqueness of Riemann solutions and the convergence of self-similar scaling sequence of solutions to show the asymptotic behavior of largeBVsolutions for the 3×3 system of Euler equations in thermoelasticity. These results indicate that the Riemann solution is the unique attractor of large discontinuous entropy solutions, whose initial data areL∞∩L1orBV∩L1perturbation of the Riemann data, for these systems. These approaches also work for proving the large-time behavior of approximate solutions to hyperbolic conservation laws.