Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic stability of the viscous shock wave is established under some smallness conditions. The proof is given by an elementary energy estimate.Comment: 20 page

Asymptotic Behavior of a Viscous Liquid-Gas Model with Mass-Dependent Viscosity and Vacuum

In this paper, we consider two classes of free boundary value problems of a viscous two-phase liquid-gas model relevant to the flow in wells and pipelines with mass-dependent viscosity coefficient. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. We obtain the asymptotic behavior and decay rates of the mass functions $n(x,t)$,\$m(x,t)$ when the initial masses are assumed to be connected to vacuum both discontinuously and continuously, which improves the corresponding result about Navier-Stokes equations in \cite{Zhu}.Comment: 24 page