Existence of quasilinear relaxation shock profiles

TL;DR

This paper proves the existence and decay rates of small-amplitude quasilinear relaxation shock profiles, extending previous methods to handle degeneracy with a Nash-Moser iteration.

Contribution

It introduces a novel application of Nash-Moser iteration for quasilinear shocks, generalizing prior semilinear approaches to degenerate cases.

Findings

Established existence of shock profiles with sharp decay rates

Extended analysis to degenerate profile ODEs

Applied Nash-Moser iteration in a new context

Abstract

We establish existence with sharp rates of decay and distance from the Chapman--Enskog approximation of small-amplitude quasilinear relaxation shocks in the general case that the profile ODE may become degenerate. Our method of analysis follows the general approach used by M\'etivier and Zumbrun in the semilinear case, based on Chapman--Enskog expansion and the macro--micro decomposition of Liu and Yu. In the quasilinear case, however, we find it necessary to apply a parameter-dependent Nash-Moser iteration to close the analysis, whereas, in the semilinear case, a simple contraction-mapping argument sufficed.