Boundary layers in weak solutions of hyperbolic conservation laws II. self-similar vanishing diffusion limits

Abstract

This paper is concerned with the boundary layers that arise in solutions of a nonlinear hyperbolic system of conservation laws in presence of vanishing diffusion. We consider self-similar solutions of the Riemann problem in a half-space, following a pioneering idea by Dafermos for the standard Riemann problem. The system is strictly hyperbolic but no assumption of genuine nonlinearity is made; moreover, the boundary is possibly characteristic, that is, the wave speed do not have a specific sign near the (stationary) boundary.<br> First, we generalize a technique due to Tzavaras and show that the boundary Riemann problem with diffusion admits a family of continuous solutions that remain uniformly bounded in the total variation norm. Careful estimates are necessary to cope with waves that collapse at the boundary and generate the boundary layer.<br> Second, we prove the convergence of these continuous solutions toward weak solutions of the Riemann problem when the diffusion parameter approaches zero. Following Dubois and LeFloch, we formulate the boundary condition in a weak form, based on a set of admissible boundary traces. Following Part I of this work, we identify and rigorously analyze the boundary set associated with the zero-diffusion method. In particular, our analysis fully justifies the use of the scaling $1/\varepsilon$ near the boundary (where $\varepsilon$ is the diffusion parameter), even in the characteristic case as advocated in Part I by the authors.