On shock waves from the inhomogeneous Boltzmann equation

TL;DR

This paper revisits shock wave structures in gases modeled by the Boltzmann equation, correcting previous self-similar solutions to ensure finite energy and physical plausibility, especially for soft potentials.

Contribution

It provides a corrected self-similar form for shock wave solutions that have finite energy, improving the physical realism of previous models.

Findings

Corrected the self-similar form for shock solutions.

Ensured solutions have finite energy.

Reduced unphysical growth of perturbations at large distances.

Abstract

We revisit the problem on the inner structure of shock waves in simple gases modelized by the Boltzmann kinetic equation. In \cite{pomeau1987shock}, a self-similarity approach was proposed for infinite total cross section resulting from a power law interaction, but this self-similar form does not have finite energy. Motivated by the work of Pomeau, Bobylev and Cercignani started the rigorous study of the solutions of the spatial homogeneous Boltzmann equation, focusing on those which do not have finite energy \cite{bobylev2002self,bobylev2003eternal}. In the present work, we provide a correction to the self-similar form, so that the solutions are more physically sound in the sense that the energy is no longer infinite and that the perturbation brought by the shock does not grow at large distances of it on the cold side in the soft potential case.