Existence and behavior of steady solutions on an interval for general hyperbolic-parabolic systems of conservation laws

TL;DR

This paper investigates the existence, uniqueness, stability, and boundary layer behavior of steady solutions for hyperbolic-parabolic conservation laws on an interval, revealing phenomena like characteristic boundary layers in the inviscid limit.

Contribution

It provides new insights into boundary layer phenomena and the behavior of solutions in small- and large-viscosity limits for general hyperbolic-parabolic systems.

Findings

Existence and stability of small amplitude solutions for symmetrizable systems.

Identification of characteristic boundary layers in the inviscid limit.

Introduction of a new type of hyperbolic boundary condition in the inviscid limit.

Abstract

We study the inflow-outflow boundary value problem on an interval, the analog of the 1D shock tube problem for gas dynamics, for general systems of hyperbolic-parabolic conservation laws. In a first set of investigations, we study existence, uniqueness, and stability, showing in particular local existence, uniqueness, and stability of small amplitude solutions for general symmetrizable systems. In a second set of investigations, we investigate structure and behavior in the small- and large-viscosity limits. A phenomenon of particular interest is the generic appearance of characteristic boundary layers in the inviscid limit, arising from noncharacteristic data for the viscous problem, even of arbitrarily small amplitude. This induces an interesting new type of \transcharacteristic" hyperbolic boundary condition governing the formal inviscid limit.