Hilbert Expansion of Boltzmann Equation with Soft Potentials and Specular Boundary Condition in Half-space

TL;DR

This paper rigorously justifies the hydrodynamic limit from the Boltzmann equation with soft potentials to the compressible Euler equations using Hilbert expansion, addressing boundary effects and boundary layer solutions.

Contribution

It introduces a novel approach to handle boundary layers in the Boltzmann equation with soft potentials, overcoming difficulties related to decay rates and collision frequency effects.

Findings

Validated the hydrodynamic limit in half-space with boundary effects.

Developed new techniques for existence of Knudsen layer solutions.

Addressed challenges posed by weak collision effects in soft potentials.

Abstract

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. We justify rigorously the validity of the hydrodynamic limit from the Boltzmann equation of soft potentials to the compressible Euler equations by the Hilbert expansion with multi-scales. Specifically, the Boltzmann solutions are expanded into three parts: interior part, viscous boundary layer and Knudsen boundary layer. Due to the weak effect of collision frequency of soft potentials, new difficulty arises when tackling the existence of Knudsen layer solutions with space decay rate, which has been overcome under some constraint conditions and losing velocity weight arguments.