Asymptotic stability of rarefaction waves for compressible Navier-Stokes equations with relaxation

TL;DR

This paper proves the long-term stability and convergence of rarefaction waves in one-dimensional relaxed compressible Navier-Stokes equations with different initial conditions, using energy methods.

Contribution

It establishes the asymptotic stability of rarefaction waves for relaxed compressible Navier-Stokes equations with varying far-field data, a novel result in this context.

Findings

Unique global solutions exist for initial data with different far-field values.

Solutions converge uniformly to rarefaction waves as time approaches infinity.

The proof utilizes $L^2$ energy methods.

Abstract

The asymptotic stability of rarefaction wave for 1-d relaxed compressible isentropic Navier-Stokes equations is established. For initial data with different far-field values, we show that there exists a unique global in time solution. Moreover, as time goes to infinity, the obtained solutions are shown to converge uniformly to rarefaction wave solution of $p$-system with corresponding Riemann initial data. The proof is based on $L^2$ energy methods.