Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations

Abstract

This paper deals with the zero dissipation limit problem for the Navier-Stokes equations when the viscosity and the heat-conductivity are of the same order.In the case when the Riemann solution of the Euler equations is piecewise constants with a contact discontinuity,we prove that there exist global solutions to the compressible Navier-Stokes equations, which converge to the in-viscid solution away from the contact discontinuity on any finite time interval, at some convergence rate as the dissipations tend towards zero.In addition, a faster convergence rate is obtained, so long as the strength of contact discontinuity $\delta=|\theta_+ -\theta_-|$is taken suitably small.