Compressible Euler limit from Boltzmann equation with Maxwell reflection boundary condition in half-space

TL;DR

This paper proves the local-in-time existence of solutions to the scaled Boltzmann equation with Maxwell reflection boundary condition in a half-space, connecting kinetic theory with compressible Euler limits and extending previous results.

Contribution

It establishes the existence of Hilbert expansion solutions under Maxwell reflection boundary conditions, extending prior work from specular to Maxwell reflection and justifying formal asymptotic analysis.

Findings

Existence of classical solutions for small Knudsen number

Extension from specular to Maxwell reflection boundary condition

Validation of formal asymptotic analysis in boundary cases

Abstract

Starting from the local-in-time classical solution to the compressible Euler system with impermeable boundary condition in half-space, by employing the coupled weak viscous layers (governed by linearized compressible Prandtl equations with Robin boundary condition) and linear kinetic boundary layers, and the analytical tools in \cite{Guo-Jang-Jiang-2010-CPAM} and some new boundary estimates both for Prandtl and Knudsen layers, we proved the local-in-time existence of Hilbert expansion type classical solutions to the scaled Boltzmann equation with Maxwell reflection boundary condition with accommodation coefficient $\alpha_\varepsilon=O(\sqrt{\varepsilon})$ when the Knudsen number $\varepsilon$ small enough. As a consequence, this justifies the corresponding case of formal analysis in Sone's books \cite{Sone-2002book, Sone-2007-Book}. This also extends the results in \cite{GHW-2020} from specular to Maxwell reflection boundary condition. Both of this paper and \cite{GHW-2020} can be viewed as generalizations of Caflisch's classic work \cite{Caflish-1980-CPAM} to the cases with boundary.