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

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.