Hyperbolic Conservation Laws

This paper is concerned with the initial-boundary value problem for a nonlinear hyper- bolic 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, Godunov scheme). We demonstrate that differ- ent regularization methods generate different boundary layers. Hence, the boundary condition can be formulated only if an approximation scheme is selected first. Assuming solely uniform L1 bounds on the approximate solutions and so dealing with L1 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 avail- able, the boundary Young measure reduces to a Dirac mass. Form the above analysis, we deduce several formulations for the boundary condition which apply whether the boundary is character- istic or not. Each formulation is based a set of admissible boundary values, following Dubois and LeFloch's terminology in "Boundary conditions for nonlinear hyperbolic systems of conservation laws", J. Diff. Equa. 71 (1988), 93-122. The local structure of those 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.