Stability of the planar rarefaction wave to two-dimensional compressible Navier-Stokes equations

TL;DR

This paper proves the nonlinear stability of planar rarefaction waves in two-dimensional compressible Navier-Stokes equations, extending known one-dimensional results to multi-dimensional systems with physical viscosities.

Contribution

It provides the first stability result for planar rarefaction waves in multi-dimensional compressible Navier-Stokes equations with physical viscosities.

Findings

Proved time-asymptotic nonlinear stability of planar rarefaction waves.

Extended stability results from 1D to 2D systems.

First such stability proof for multi-dimensional systems with physical viscosities.

Abstract

It is well-known that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws, is nonlinearly stable to the one-dimensional compressible Navier-Stokes equations (cf. [14,15,12,17]). In the present paper we proved the time-asymptotically nonlinear stability of the planar rarefaction wave to the two-dimensional compressible and isentropic Navier-Stokes equations, which gives the first stability result of the planar rarefaction wave to the multi-dimensional system with physical viscosities.