Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit

TL;DR

This paper studies the thermalization process of a rarefied gas conserving total energy, proving existence and stability of solutions, and deriving the macroscopic limit to a cross-diffusion model.

Contribution

It establishes existence and hypocoercivity results for a coupled kinetic and heat system with energy conservation, and formally derives its macroscopic limit.

Findings

Existence of solutions on the 1D torus.

Spectral stability via hypocoercivity in arbitrary dimensions.

Formal derivation of the macroscopic cross-diffusion limit.

Abstract

The thermalization of a gas towards a Maxwellian velocity distribution with the background temperature is described by a kinetic relaxation model. The sum of the kinetic energy of the gas and the thermal energy of the background are conserved, and the heat flow in the background is governed by the Fourier law. For the coupled nonlinear system of the kinetic and the heat equation, existence of solutions is proved on the one-dimensional torus. Spectral stability of the equilibrium is shown on the torus in arbitrary dimensions by hypocoercivity methods. The macroscopic limit towards a nonlinear cross-diffusion problem is carried out formally.