Boundary layers of the Boltzmann equation in a three-dimensional half-space

TL;DR

This paper proves the existence and asymptotic stability of stationary boundary layer solutions for the nonlinear Boltzmann equation in a three-dimensional half-space, extending previous work from one-dimensional models.

Contribution

It establishes the unique existence and stability of boundary layer solutions for the 3D Boltzmann equation, a significant advancement over prior 1D analyses.

Findings

Existence of stationary solutions in 3D half-space

Asymptotic stability of these solutions

Extension of boundary layer analysis to three dimensions

Abstract

We consider the nonlinear boundary layers of the Boltzmann equation in a three-dimensional half-space by perturbing around a Maxwellian, under the assumption that the Mach number of the Maxwellian satisfies ${\cal M}_{\infty} < -1$. In preceding works, nonlinear boundary layers of the Boltzmann equation in a half-line are considered, with stationary solutions obtained and nonlinear stability confirmed. In this paper, we establish the unique existence of stationary solutions for the three-dimensional half-space model, and show that the stationary solution is asymptotic stable.