Boundary layers in weak solutions of hyperbolic conservation laws

Abstract

This paper is concerned with the initial-boundary-value problem for a nonlinear hyperbolic system of conservation laws. We study the boundary layers that may arise in approximations of entropy discontinuous solutions. We consider both the vanishing-viscosity method and finite-difference schemes (Lax-Friedrichs-type schemes and the Godunov scheme). We demonstrate that different regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform bounds on the approximate solutions and so dealing with solutions, we derive several entropy inequalities satisfied by the boundary layer in each case under consideration. A Young measure is introduced to describe the boundary trace. When a uniform bound on the total variation is available, the boundary Young measure reduces to a Dirac mass. From the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is characteristic or not. Each formulation is based on a set of admissible boundary values, following the terminology of Dubois and LeFloch. The local structure of these sets and the well-posedness of the corresponding initial-boundary-value problem are investigated. The results are illustrated with convex and nonconvex conservation laws and examples from continuum mechanics.