Asymptotic stability of composite waves of viscous shock and rarefaction for relaxed compressible Navier-Stokes equations

TL;DR

This paper proves the long-term stability of combined shock and rarefaction waves in relaxed compressible Navier-Stokes equations, showing convergence to classical solutions as relaxation vanishes, using entropy and energy methods.

Contribution

It establishes the asymptotic nonlinear stability of composite waves in relaxed systems and their convergence to classical solutions as the relaxation parameter approaches zero.

Findings

Composite waves are asymptotically stable under small perturbations.

Solutions converge to classical system solutions as relaxation parameter tends to zero.

The methods involve relative entropy, a-contraction, and energy estimates.

Abstract

The time asymptotic stability for one-dimensional relaxed compressible Navier-Stokes equations is studied. We show that the composite waves of viscous shock and rarefaction are asymptotically nonlinear stable with both small wave strength and small initial perturbations. Moreover, as the relaxation parameter goes to zero, the solutions of relaxed system are shown to converge globally in time to that of classical system. The methods are based on relative entropy, the a-contraction with shifts theory and basic energy estimates.