BGK model for rarefied gas in a bounded domain

TL;DR

This paper proves the existence, uniqueness, and exponential stability of solutions to the BGK equation in bounded domains with diffusive boundary conditions, using linearized analysis and nonlinear estimates.

Contribution

It establishes the global well-posedness and stability of the BGK model in bounded domains with general collision frequency, extending previous results to more general boundary conditions.

Findings

Unique global solution with exponential convergence rate

Established $L^2$ coercive estimate for linearized operator

Derived $L^ Infty$ bounds for nonlinear stability

Abstract

We study the Bathnagar-Gross-Krook (BGK) equation in a smooth bounded domain featuring a diffusive reflection boundary condition with general collision frequency. We prove that the BGK equation admits a unique global solution with an exponential convergence rate if the initial condition is a small perturbation around the global Maxwellian in the $L^\infty$ space. For the proof, we utilize the dissipative nature from the linearized BGK operator and establish an $L^2$ coercive estimate. Next, we derive the a priori estimate by obtaining an $L^\infty$ bound on the nonlinear operator; this requires a delicate analysis to manage its intrinsic nonlinear structure. Finally, we establish the $L^\infty$ stability estimate and introduce sequential arguments for the nonlinear BGK operator, thereby concluding both well-posedness and positivity.