The stability of strong viscous contact discontinuity to a free boundary problem for compressible Navier-Stokes equations

TL;DR

This paper proves the nonlinear global stability of viscous contact discontinuities in a free boundary problem for the compressible Navier-Stokes equations, even with large temperature differences and big perturbations.

Contribution

It establishes the global nonlinear stability of viscous contact discontinuities with large temperature oscillations in a free boundary setting, extending previous local stability results.

Findings

Global stability holds for large temperature differences.

Stability is achieved under certain large perturbations.

Energy methods are used to prove the stability.

Abstract

This paper is concerned with nonlinear stability of viscous contact discontinuity to a free boundary problem for the one-dimensional full compressible Navier-Stokes equations in half space $[0,\infty)$. For the case when the local stability of the contact discontinuities was first studied by [1],later generalized by [2], local stability of weak viscous contact discontinuity is well-established by [4-8], but for the global stability of the impermeable gas with big oscillation ends $(|\theta_+-\theta_-|>1) $, fewer results have been obtained excluding zero dissipation [9] or $\gamma\to 1$ gas see [10]. Our main purpose is to deduce the corresponding nonlinear stability result with the two different ends to temperature by exploiting the elementary energy method. As a first step towards this goal, we will show in this paper that with a certain class of big perturbation which can allow $|\theta_--\theta_+|>1$, the global stability result holds.