Hydrodynamic limit of the Boltzmann equation to the planar rarefaction wave in three dimensional space

TL;DR

This paper proves the global hydrodynamic limit of the Boltzmann equation to a planar rarefaction wave in three dimensions, using advanced analytical techniques to handle large wave strengths and localized estimates.

Contribution

It introduces a generalized Hilbert expansion and an $L^2-L^ abla$ framework to establish the limit for general collision kernels in 3D.

Findings

Established global hydrodynamic limit for Boltzmann to rarefaction wave

Improved $L^2$ estimates with localization for 3D problem

Handled large wave strength in the analysis

Abstract

In this paper, we establish the global in time hydrodynamic limit of Boltzmann equation to the planar rarefaction wave of compressible Euler system in three dimensional space $x\in\mathbb{R}^3$ for general collision kernels. Our approch is based on a generalized Hilbert expansion, and a recent $L^2-L^\infty$ framework. In particular, we improve the $L^2$-estimate to be a localized version because the planar rarefaction wave is indeed a one-dimensional wave which makes the source terms to be not integrable in the $L^2$ energy estimate of three dimensional problem. We also point out that the wave strength of rarefaction may be large.