Stability of scalar radiative shock profiles

TL;DR

This paper proves the nonlinear orbital asymptotic stability of scalar radiative shock profiles, which are traveling wave solutions in a simplified radiating gas model, using Green function bounds and resolvent kernel construction.

Contribution

It introduces a novel analysis method for degenerate eigenvalue systems and establishes stability results for scalar radiative shock profiles.

Findings

Proved nonlinear orbital asymptotic stability of scalar radiative shock profiles.

Developed new resolvent kernel construction for degenerate eigenvalue systems.

Established pointwise Green function bounds for the linearized operator.

Abstract

This work establishes nonlinear orbital asymptotic stability of scalar radiative shock profiles, namely, traveling wave solutions to the simplified model system of radiating gas \cite{Hm}, consisting of a scalar conservation law coupled with an elliptic equation for the radiation flux. The method is based on the derivation of pointwise Green function bounds and description of the linearized solution operator. A new feature in the present analysis is the construction of the resolvent kernel for 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.