Stability of radiative shock profiles for hyperbolic-elliptic coupled systems

TL;DR

This paper proves the nonlinear orbital stability of small-amplitude radiative shock profiles in coupled hyperbolic-elliptic systems modeling radiative gases, extending previous scalar results to more general systems.

Contribution

It introduces a new method for establishing stability of shock profiles in coupled hyperbolic-elliptic systems, including the Euler-Poisson model, using Green function bounds and resolvent kernel construction.

Findings

Proved nonlinear orbital stability of small-amplitude shock profiles.

Developed pointwise Green function bounds for the linearized operator.

Constructed the resolvent kernel for degenerate eigenvalue systems.

Abstract

Extending previous work with Lattanzio and Mascia on the scalar (in fluid-dynamical variables) Hamer model for a radiative gas, we show nonlinear orbital asymptotic stability of small-amplitude shock profiles of general systems of coupled hyperbolic--eliptic equations of the type modeling a radiative gas, that is, systems of conservation laws coupled with an elliptic equation for the radiation flux, including in particular the standard Euler--Poisson model for a radiating gas. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator, with the main difficulty being the construction of the resolvent kernel in the case of an eigenvalue system of equations of degenerate type. Nonlinear stability then follows in standard fashion by linear estimates derived from these pointwise bounds, combined with nonlinear-damping type energy estimates