Boundary conditions and boundary layers for a multidimensional relaxation model

In this paper we study the linearized relaxation model of Katsoulakis and Tzavaras in a half-space with arbitrary space dimension n>=1. Our main interest is to establish the asymptotic equivalence of the relaxation system and its corresponding multi-dimensional equilibrium conservation law. We identify and rigorously justify a necessary and sufficient condition (which we refer to as stiff Kreiss condition, or SKC in short) on the boundary condition to guarantee the uniform stability of the initial– boundary value problem of the relaxation system independent of the relaxation rate. The asymptotic convergence and the corresponding boundary layer behavior are studied by Fourier–Laplace transform and a detailed asymptotic analysis. The SKC is shown to be more restrictive than the classical uniform Kreiss condition for all n>=1.