Shock profiles for the non-cutoff Boltzmann equation with hard potentials

TL;DR

This paper constructs smooth shock profile solutions for the non-cutoff Boltzmann equation with long-range interactions, demonstrating their approximation by classical Navier-Stokes shock profiles using energy estimates and Chapman-Enskog methods.

Contribution

It extends the analysis of shock profiles to the non-cutoff Boltzmann equation with long-range potentials, a case not previously addressed.

Findings

Shock profiles are smooth and well approximated by Navier-Stokes solutions.

The construction uses energy estimates and Chapman-Enskog approximation.

Results advance understanding of gas dynamics with long-range molecular interactions.

Abstract

The Boltzmann equation models gas dynamics in the low density or high Mach number regime, using a statistical description of molecular interactions. Planar shock wave solutions have been constructed for the Boltzmann equation for hard potentials with angular cutoff, and more recently for the Landau equation of plasma dynamics. In this work, we construct shock profile solutions for the Boltzmann equation where the molecular interactions are long-range, and we show these solutions to be smooth and well approximated by compressible Navier Stokes shock profiles. Our proof procedes by standard energy estimates and a quantitative Chapman-Enskog approximation.