Existence and sharp localization in velocity of small-amplitude Boltzmann shocks

TL;DR

This paper proves the existence of small-amplitude Boltzmann shock profiles with sharp decay rates and velocity localization, using a weighted contraction mapping approach and analyzing the negative definiteness of the linearized collision operator.

Contribution

It introduces an elementary proof of shock profile existence with improved decay and localization rates by exploring the negative definiteness of the linearized collision operator in broader norms.

Findings

Established existence of Boltzmann shock profiles with sharp decay rates.

Demonstrated negative definiteness of the linearized collision operator in wider norms.

Achieved sharper velocity localization rates near Maxwellian.

Abstract

Using a weighted $H^s$-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman--Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere potential, recovering and slightly sharpening results obtained by Caflisch and Nicolaenko using different techniques. A key technical point in both analyses is that the linearized collision operator $L$ is negative definite on its range, not only in the standard square-root Maxwellian weighted norm for which it is self-adjoint, but also in norms with nearby weights. Exploring this issue further, we show that $L$ is negative definite on its range in a much wider class of norms including norms with weights asymptotic nearly to a full Maxwellian rather than its square root. This yields sharp localization in velocity at near-Maxwellian rate, rather than the square-root rate obtained in previous analyses