Boundary layer solution of the Boltzmann equation for specular boundary condition

TL;DR

This paper proves the existence, uniqueness, and decay properties of steady boundary layer solutions to the Boltzmann equation with specular boundary conditions in a half-space, which are crucial for the Hilbert expansion analysis.

Contribution

It establishes the first rigorous existence and decay results for boundary layer solutions of the Boltzmann equation with specular boundary conditions in half-space.

Findings

Existence of steady boundary layer solutions in $L^2_{x,v} igcap L^ abla_{x,v}$.

Solutions exhibit exponential decay and are unique and continuous.

Results facilitate the proof of Hilbert expansion for half-space Boltzmann problems.

Abstract

In the paper, we establish the existence of steady boundary layer solution of Boltzmann equation with specular boundary condition in $L^2_{x,v}\cap L^\infty_{x,v}$ in half-space. The uniqueness, continuity and exponential decay of the solution are obtained, and such estimates are important to prove the Hilbert expansion of Boltzmann equation for half-space problem with specular boundary condition.