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September  2013, 12(5): 1907-1926. doi: 10.3934/cpaa.2013.12.1907

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

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

Received  July 2011 Revised  November 2012 Published  January 2013

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.
Citation: Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907
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Anal., 44 (2012), 1742. doi: 10.1137/100814305. Google Scholar [5] F. Huang, A. Matsumura and Z. Xin, Stability of contact Discontinuities for the 1-D compressible Navier-Stokes equations,, Arch. Ration. Mech. Anal., 179 (2005), 55. doi: 10.1007/s00205-005-0380-7. Google Scholar [6] F. Huang, Y. Wang and T. Yang, Fluid dynamic limit to the Riemann solutions of Euler equations: I. Superposition of rarefaction waves and contact dis-continuity,, Kinetic and Related Models, 3 (2010), 685. doi: 10.3934/krm.2010.3.685. Google Scholar [7] S. Jiang, G. Ni and W. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equations of one-dimensional compressible heat-conducting fluids,, SIAM J. Math. Anal., 38 (2006), 368. doi: 10.1137/050626478. Google Scholar [8] S. Kawashima, Large-time behaviour of solutions to hyperbolic–parabolic systems of conservation laws and applications,, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169. doi: 10.1017/S0308210500018308. 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Eqns., 249 (2010), 827.  doi: 10.1016/j.jde.2010.03.011.  Google Scholar [15] Z. Xin, On nonlinear stability of contact discontinuities,, In, (1994), 249.   Google Scholar [16] Z. Xin, Zero dissipation limit to rarefaction waves for the 1-dimensional Navier-Stokes equations of compressible isentropic gases,, Comm. Pure. Appl. Math., 46 (1993), 621.  doi: 0010-3640/93/050621-45.  Google Scholar
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