Hilbert expansion of the Boltzmann equation in the incompressible Euler level in a channel

TL;DR

This paper rigorously derives the incompressible Euler equations from the Boltzmann equation in a channel, accounting for boundary layers and using multiscale Hilbert expansions to justify the hydrodynamic limit.

Contribution

It develops a multiscale Hilbert expansion framework to handle boundary layers and rigorously justify the hydrodynamic limit from Boltzmann to Euler equations in a bounded domain.

Findings

Established uniform estimates for solutions and boundary layers.

Constructed solutions via truncated multiscale Hilbert expansion.

Justified the hydrodynamic limit in the incompressible Euler regime.

Abstract

The study of hydrodynamic limit of the Boltzmann equation with physical boundary is a challenging problem due to appearance of the viscous and Knudsen boundary layers. In this paper, the hydrodynamic limit from the Boltzmann equation with specular reflection boundary condition to the incompressible Euler in a channel is investigated. Based on the multiscaled Hilbert expansion, the equations with boundary conditions and compatibility conditions for interior solutions, viscous and Knudsen boundary layers are derived under different scaling, respectively. Then some uniform estimates for the interior solutions, viscous and Knudsen boundary layers are established. With the help of $L^2-L^\infty$ framework and the uniform estimates obtained above, the solutions to the Boltzmann equation are constructed by the truncated Hilbert expansion with multiscales, and hence the hydrodynamic limit in the incompressible Euler level is justified.