Stability of Rarefaction Waves to the 1D Compressible Navier-Stokes Equations with Density-dependent Viscosity

TL;DR

This paper proves the stability of rarefaction waves in the 1D compressible Navier-Stokes equations with density-dependent viscosity, showing vacuum states vanish and solutions become strong over time.

Contribution

It establishes the weak stability of rarefaction waves for large amplitudes and arbitrary initial perturbations with vacuum states, using a novel approximation and regularization approach.

Findings

Vacuum states vanish in finite time

Weak solutions become unique strong solutions

Stability holds for large-amplitude waves

Abstract

In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is constructed by approximating the system and regularizing the initial values which may contain vacuum state. Then some global in time estimates on the weak solution are obtained. Based on these uniform estimates, the vacuum states are shown to vanish in finite time and the weak solution we constructed becomes a unique strong one. Consequently, the stability of the rarefaction wave is proved in a weak sense. The theory holds for large-amplitudes rarefaction waves and arbitrary initial perturbations.