Boundary layers in weak solutions of hyperbolic conservation laws. III. Vanishing relaxation limits

Abstract

This is the third part of a series concerned with boundary layers in solutions of nonlinear hyperbolic systems of conservation laws. We consider here self-similar solutions of the Riemann problem, following a pioneering idea by Dafermos. The system under study is strictly hyperbolic but no assumption of genuine nonlinearity is made. The boundary is possibly characteristic, that the sign of the characteristic speed near the boundary is not known a priori. We investigate the effect of vanishing relaxation terms on the solutions of the Riemann problem. We show that the boundary Riemann problem with relaxation admits continuous solutions that remain uniformly bounded in the total variation norm. Following the second part of this series, we derive the necessary uniform estimates near the boundary which allow us to describe the structure of the boundary layer even when the boundary is characteristic. Our analysis provides still a new approach to the existence of Riemann solutions for systems of conservation laws.