Stability of planar rarefaction wave to 3D full compressible Navier-Stokes equations

TL;DR

This paper proves the stability of planar rarefaction waves in three-dimensional full compressible Navier-Stokes equations within an infinite nozzle, highlighting new cancellation techniques to handle wave interactions in multiple directions.

Contribution

It introduces novel cancellation observations for flux and viscous terms, enabling the extension of stability results from 1D to 3D with complex wave interactions.

Findings

Established time-asymptotic stability of planar rarefaction waves in 3D

Developed new analytical techniques for wave cancellations in multiple dimensions

Extended 1D stability results to 3D full compressible Navier-Stokes equations

Abstract

We prove the time-asymptotic stability toward planar rarefaction wave for the three-dimensional full compressible Navier-Stokes equations in an infinite long flat nozzle domain $\mathbb{R}\times\mathbb{T}^2$. Compared with one-dimensional case, the proof here is based on our new observations on the cancellations on the flux terms and viscous terms due to the underlying wave structures, which are crucial to overcome the difficulties due to the wave propagation along the transverse directions $x_2$ and $x_3$ and its interactions with the planar rarefaction wave in $x_1$ direction.