Periodic perturbations of a composite wave of two viscous shocks for 1-D full compressible Navier-Stokes equations

TL;DR

This paper proves the asymptotic stability of a composite wave of two viscous shocks in the 1-D full compressible Navier-Stokes equations under periodic perturbations, showing the solution converges to a shifted wave over time.

Contribution

It introduces a novel approach using a suitable ansatz and the anti-derivative method to establish stability under periodic perturbations.

Findings

Solution approaches the background wave with a shift over time

Shifts are uniquely determined for small perturbations and shock strengths

Method can handle spatially periodic perturbations in viscous shock analysis

Abstract

This paper is concerned with the asymptotic stability of a composite wave of two viscous shocks under spatially periodic perturbations for the 1-D full compressible Navier-Stokes equations. It is proved that as time increases, the solution approaches the background composite wave with a shift for each shock, where the shifts can be uniquely determined if both the periodic perturbations and strengths of two shocks are small. The key of the proof is to construct a suitable ansatz such that the anti-derivative method works.