Hilbert expansion of the Boltzmann equation with specular boundary condition in half-space

TL;DR

This paper rigorously justifies the Hilbert expansion for the Boltzmann equation with specular boundary conditions, connecting kinetic theory to hydrodynamic limits like Euler and acoustic equations.

Contribution

It provides a systematic derivation and validation of the Hilbert expansion in the presence of boundary effects, specifically for specular reflection in half-space.

Findings

Validates the Hilbert expansion with boundary effects

Derives compressible Euler equations from Boltzmann equation

Establishes connection between kinetic and hydrodynamic models

Abstract

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic derivation and study of the viscous layer equations and the $L^2$ to $L^\infty$ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations.