Compressible Euler limit from Boltzmann equation with complete diffusive boundary condition in half-space

TL;DR

This paper rigorously derives the compressible Euler equations as a limit of the Boltzmann equation with diffusive boundary conditions in a half-space, using Hilbert expansion and layers, extending previous results to boundary problems.

Contribution

It provides a rigorous proof of the Euler limit from Boltzmann with diffusive boundary conditions, including interior and Knudsen layers, for short time smooth solutions.

Findings

Justifies formal analysis in Sone's book

Generalizes Caflisch's result to boundary problems

Establishes the Euler limit in half-space with diffusive boundary

Abstract

In this paper, we prove the compressible Euler limit from the Boltzmann equation with hard sphere collisional kernel and complete diffusive boundary condition in half-space by employing the Hilbert expansion which includes interior and Knudsen layers. This rigorously justifies the corresponding formal analysis in Sone's book \cite{Sone-2007-Book} in the context of short time smooth solutions, and also generalizes the classic Caflisch's result \cite{Caflish-1980-CPAM} to initial-boundary problem case.