On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary

Abstract

where ( ) is the velocity, ρ( ) > 0 the density, θ( ) the absolute temperature, μ > 0 the viscosity constant and κ > 0 the coefficient of heat conduction. The pressure = (ρ θ) and the internal energy = (ρ θ) are related by the second law of thermodynamics. There have been a lot of works on the asymptotic behaviors of the solutions for the system (1.1). Most of these results are concerned with the rarefaction wave and viscous shock wave. We refer to [10–15] for 2 × 2 case and [4–5, 7–8] for 3 × 3 case and references therein. However there is no result on the contact discontinuity for the system (1.1) until now due to various difficulties. Although some progress on the contact discontinuity were obtained by Liu and Xin [9] and Xin [17] in which the asymptotic toward the contact discontinuity was investigated for the initial value problem (IVP) of viscous conservation laws with uniformly artificial viscosity, no result is known for the physical system, especially for the compressible N-S equations (1.1). Therefore we really want to give a positive result on the contact discontinuity for the physical system (1.1). To simplify our problem, we focus our attention on the perfect gas. In this situation,