Boundary Conditions for Hyperbolic Relaxation Systems with Characteristic Boundaries of Type I

TL;DR

This paper develops a modified boundary condition framework for hyperbolic relaxation systems with characteristic boundaries, extending existing theories and verifying stability and boundary-layer phenomena.

Contribution

It introduces a modified Generalized Kreiss condition for characteristic boundaries and demonstrates its effectiveness in stability analysis and boundary-layer existence.

Findings

Extended GKC to characteristic boundaries.

Validated boundary condition stability via energy estimates.

Proved existence of boundary-layers in nonlinear cases.

Abstract

This work is concerned with boundary conditions for one-dimensional hyperbolic relaxation systems with characteristic boundaries. We assume that the relaxation system satisfies the structural stability condition proposed by the second author previously and the boundary is characteristic of type I (characteristic for the relaxation system but non-characteristic for the corresponding equilibrium system). For this kind of characteristic initial-boundary-value problems, we propose a modified Generalized Kreiss condition (GKC). This extends the GKC proposed by the second author for the non-characteristic boundaries to the present characteristic case. Under this modified GKC, we derive the reduced boundary condition and verify its validity by combining an energy estimate with the Laplace transform. Moreover, we show the existence of boundary-layers for nonlinear problems.