Cauchy problem for the Boltzmann-BGK model near a global Maxwellian

TL;DR

This paper proves the existence, uniqueness, and stability of global smooth solutions for the Boltzmann-BGK model near a Maxwellian, including decay estimates, under small initial perturbations.

Contribution

It establishes the global well-posedness and decay properties of solutions to the Boltzmann-BGK model with general collision frequencies near a Maxwellian.

Findings

Existence of unique global smooth solutions

Asymptotic decay estimates proven

Uniform L2-stability for nonlinear perturbations

Abstract

In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK model for a general class of collision frequencies. We prove that the Boltzmann-BGK model linearized around a global Maxwellian admits a unique global smooth solution if the initial perturbation is sufficiently small in a high order energy norm. We also establish an asymptotic decay estimate and uniform $L^2$-stability for nonlinear perturbations.