Boundary Layer Solution of the Boltzmann Equation for Diffusive Reflection Boundary Conditions in Half-space

TL;DR

This paper establishes the existence, continuity, decay, and uniqueness of boundary layer solutions for the steady Boltzmann equation with diffusive reflection boundary conditions in a half-space, under specific conditions.

Contribution

It provides the first rigorous proof of boundary layer solutions for both linear and nonlinear Boltzmann equations with diffusive reflection in half-space.

Findings

Existence of boundary layer solutions under certain conditions

Solutions exhibit spatial decay and continuity

Uniqueness established under constraints

Abstract

We study steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions and the source term decaying exponentially, we establish the existence of boundary layer solution for both linear and nonlinear Boltzmann equation in half-space with diffusive reflection boundary condition in $L^\infty_{x,v}$ when the far-field Mach number of the Maxwellian is zero. The continuity and the spacial decay of the solution are obtained. The uniqueness is established under some constraint conditions.