On the stability of contact discontinuity for Cauchy problem of compress Navier-Stokes equations with general initial data

Abstract

In this paper, we study the stability of solutions of the Cauchy problem for 1-D compressible Narvier-Stokes equations with general initial data. The asymptotic limit of solution is found, under some conditions. The results in this paper imply the case that the limit function of solution as t -> a is a viscous contact wave in the sense, which approximates the contact discontinuity on any finite-time interval as the heat conduction coefficients toward zero. As a by-product, the decay rates of the solution for the fast diffusion equations are also obtained. The proofs are based on the elementary energy method and the study of asymptotic behavior of the solution to the fast diffusion equation.