Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity

TL;DR

This paper investigates the long-term behavior of solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity near rarefaction waves connected to vacuum, showing convergence to the wave and persistence of vacuum states over time.

Contribution

It constructs global weak solutions around rarefaction waves with vacuum and proves their convergence and regularity, highlighting differences from non-vacuum cases.

Findings

Weak solutions converge to the rarefaction wave connected to vacuum.

Vacuum states persist for all time in the solution.

Solutions become regular away from vacuum regions.

Abstract

In this paper, we study the large time asymptotic behavior toward rarefaction waves for solutions to the 1-dimensional compressible Navier-Stokes equations with density-dependent viscosities for general initial data whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. First, a global-in-time weak solution around the rarefaction wave is constructed by approximating the system and regularizing the initial data with general perturbations, and some a priori uniform-in-time estimates for the energy and entropy are obtained. Then it is shown that the density of any weak solution satisfying the natural energy and entropy estimates will converge to the rarefaction wave connected to vacuum with arbitrary strength in super-norm time-asymptotically. Our results imply, in particular, that the initial vacuum at far fields will remain for all the time which are in contrast to the case of non-vacuum rarefaction waves studied in \cite{JWX} where all the possible vacuum states will vanish in finite time. Finally, it is proved that the weak solution becomes regular away from the vacuum region of the rarefaction wave.