Existence of semilinear relaxation shocks

TL;DR

This paper proves the existence and decay rates of small-amplitude shock profiles in semilinear relaxation systems, including kinetic models, using a macro-micro decomposition approach.

Contribution

It introduces a novel application of macro-micro decomposition to stationary equations, providing a simplified proof of shock profile existence.

Findings

Established existence of shock profiles with sharp decay rates.

Applied macro-micro decomposition to stationary equations.

Provided a contraction mapping proof in weighted Sobolev spaces.

Abstract

We establish existence with sharp rates of decay and distance from the Chapman--Enskog approximation of small-amplitude shock profiles of a class of semilinear relaxation systems including discrete velocity models obtained from Boltzmann and other kinetic equations. Our method of analysis is based on the macro--micro decomposition introduced by Liu and Yu for the study of Boltzmann profiles, but applied to the stationary rather than the time-evolutionary equations. This yields a simple proof by contraction mapping in weighted $H^s$ spaces.