Nonlinear stability of planar rarefaction wave to the three-dimensional Boltzmann equation

TL;DR

This paper proves the nonlinear stability of planar rarefaction waves in the three-dimensional Boltzmann equation, addressing complex wave interactions and establishing the first such result for this setting.

Contribution

It introduces a novel stability analysis for planar rarefaction waves in 3D Boltzmann equations, overcoming challenges from wave propagation and interactions.

Findings

First stability result for 3D Boltzmann rarefaction waves

Overcomes difficulties from wave interactions in transverse directions

Uses micro-macro decomposition and new wave structure insights

Abstract

We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation, based on the micro-macro decomposition introduced in [24, 22] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of planar rarefaction wave for 3D Boltzmann equation, while the corresponding results for the shock and contact discontinuities are still completely open.