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

Zhilei Liang

doi:10.3934/cpaa.2013.12.1907

Article Contents

2013, Volume 12, Issue 5: 1907-1926. Doi: 10.3934/cpaa.2013.12.1907

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

Zhilei Liang^1,

1.
School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130

Received: June 30, 2011

Revised: October 31, 2012

Published: August 2013

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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.

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Mathematics Subject Classification: Primary: 35Q30,76N15; Secondary: 35L65.

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