Global Solutions of the Boltzmann Equation over $\mathbb{R}^D$ near Global Maxwellians with Small Mass

TL;DR

This paper analyzes the long-term behavior of solutions to the Boltzmann equation in Euclidean space near Maxwellian states with small mass, revealing that solutions tend to freely transported states over time.

Contribution

It establishes the global existence and asymptotic behavior of solutions near Maxwellians with small mass in bf6n space, using scattering theory techniques.

Findings

Solutions tend to free transport states as time progresses.

The dissipative effect of collisions is quenched by dispersion.

The analysis applies to the Euclidean space setting.

Abstract

We study the dynamics defined by the Boltzmann equation set in the Euclidean space $\mathbb{R}^D$ in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.