Asymptotic stability of the combination of a viscous contact wave with two rarefaction waves for 1-D Navier-Stokes equations under periodic perturbations

TL;DR

This paper proves the asymptotic stability of a composite wave consisting of a viscous contact wave and two rarefaction waves for 1-D compressible Navier-Stokes equations under space-periodic perturbations that oscillate at the far field.

Contribution

It introduces a novel ansatz and weight function approach to handle oscillating, non-integrable perturbations in the stability analysis of composite waves.

Findings

The composite wave remains stable under periodic perturbations.

A unique global-in-time solution exists for the Cauchy problem.

The method can be applied to contact discontinuity and composite waves.

Abstract

Considering the space-periodic perturbations, we prove the time-asymptotic stability of the composite wave of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in this paper. This kind of perturbations keep oscillating at the far field and are not integrable. The key is to construct a suitable ansatz carrying the same oscillation %eliminating the oscillation of the solution as in \cite{HuangXuYuan2020,HuangYuan2021}, but due to the degeneration of contact discontinuity, the construction is more subtle. We find a way to use the same weight function for different variables and wave patterns, which still ensure the errors be controllable. Thus, this construction can be applied to contact discontinuity and composite waves. Finally, by the energy method, we prove that the Cauchy problem admits a unique global-in-time solution and the composite wave is still stable under the space-periodic perturbations.