Stability of composite Wave of Planar Viscous Shock and Rarefaction for 3D Barotropic Navier-Stokes Equations

TL;DR

This paper proves the nonlinear stability and convergence of a composite wave, combining a viscous shock and a rarefaction wave, for 3D barotropic Navier-Stokes equations under small perturbations, without zero-mass restrictions.

Contribution

It establishes the nonlinear stability of the composite wave in 3D Navier-Stokes equations using the $a$-contraction method with time-dependent shifts and weights.

Findings

Global-in-time strong solution exists for small initial perturbations.

Solution converges to the composite wave asymptotically.

Stability holds without zero-mass conditions.

Abstract

We prove the nonlinear time-asymptotic stability of the composite wave consisting of a planar rarefaction wave and a planar viscous shock for the three-dimensional (3D) compressible barotropic Navier-Stokes equations under generic perturbations, in particular, without zero-mass conditions. It is shown that if the composite wave strength and the initial perturbations are suitably small, then 3D Navier-Stokes system admits a unique global-in-time strong solution which time-asymptotically converges to the corresponding composite wave up to a time-dependent shift for planar viscous shock. Our proof is based on the $a$-contraction method with time-dependent shift and suitable weight function.