The stability of strong viscous contact discontinutiy to an inflow problem for full compressible Navier-Stokes equations

TL;DR

This paper proves the nonlinear global stability of viscous contact discontinuities in the full compressible Navier-Stokes equations with inflow boundary conditions, even for large initial perturbations with significant temperature and density differences.

Contribution

It establishes the global nonlinear stability of viscous contact discontinuities for inflow problems with large oscillations, extending previous local stability results using energy methods.

Findings

Global stability holds for large perturbations with significant temperature and density differences.

The energy method effectively proves stability without zero dissipation or gamma approaching 1.

The results apply to the full compressible Navier-Stokes equations in a half-space setting.

Abstract

This paper is concerned with nonlinear stability of viscous contact discontinuity to inflow problem for the one-dimensional full compressible Navier-Stokes equations with different ends in half space $[0,\infty)$. For the case when the local stability of the contact discontinuities was first studied by \cite{X},later generalized by \cite{LX}, local stability of weak viscous contact discontinuity is well-established by \cite{HMS,HMX,HXY,HZ,HLM2009}, but for the global stability of inflow gas with big oscillation ends $(|\theta_+-\theta_-|>1\ and \ |\rho_+-\rho_-|>1)$, fewer results have been obtained excluding zero dissipation \cite{MaSX} or $\gamma\to 1$ gas see \cite{HH}. Our main purpose is to deduce the corresponding nonlinear stability result with the two different ends 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 $|\theta_--\theta_+|>1$ and $|\rho_--\rho_+|>1$ ,the global stability result holds.