Boundary conditions for hyperbolic relaxation systems with characteristic boundaries

TL;DR

This paper investigates boundary conditions for multi-dimensional hyperbolic relaxation systems with characteristic boundaries, redefining conditions necessary for well-posedness and deriving reduced boundary conditions without assuming non-characteristic boundaries.

Contribution

It introduces a generalized Kreiss condition for characteristic boundaries and derives reduced boundary conditions for the relaxation limit, extending previous results to characteristic boundary cases.

Findings

Redefines a Generalized Kreiss condition for characteristic boundaries

Derives reduced boundary conditions for the relaxation limit

Completes the derivation of boundary conditions for general linear relaxation systems

Abstract

This paper is concerned with initial-boundary-value problems of general multi-dimensional hyperbolic relaxation systems with characteristic boundaries. For the characteristic case, we redefine a Generalized Kreiss condition (GKC) which is essentially necessary to have a well-behaved relaxation limit. Under this characteristic GKC and a Shizuta-Kawashima-like condition, we derive reduced boundary conditions for the relaxation limit solving the corresponding equilibrium systems and justify the validity thereof. The key of the derivation is to select an elaborate version of the characteristic GKC by invoking the Shizuta-Kawashima-like condition. In contrast to the existing results, the present one does not assume that the boundary is non-characteristic for either the relaxation or equilibrium systems. In this sense, this paper completes the task in deriving reduced boundary conditions for general linear relaxation systems satisfying the structural stability condition.