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

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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

Zhilei Liang ^1,

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

Received July 2011 Revised November 2012 Published January 2013

Abstract
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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.

Keywords: contact discontinuity, Euler equations, Compressible Navier-Stokes, zero dissipation limit, convergence rate..

Mathematics Subject Classification: Primary: 35Q30,76N15; Secondary: 35L6.

Citation: Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

References:

[1]	C. Duyn and L. Peletier, A class of similarity solution of the nonlinear diffusion equation, Nonlinear Analysis T.M.A., 1 (1977), 223-233. doi: 10.1016/0362-546X(77)90032-3.
[2]	J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121 (1992), 235-265. doi: 10.1007/BF00410614.
[3]	D. Hoff and T. Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentripic flow with shock data, Indiana Univ. Math. J., 38 (1989), 861-915.
[4]	F. Huang, M. Li and Y. Wang, Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations, SIAM J. Math. Anal., 44(3) (2012), 1742-1759. arXiv:1011.1991v1. doi: 10.1137/100814305.
[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), 55-77. doi: 10.1007/s00205-005-0380-7.
[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), 685-728. arXiv:1011.1990v1 doi: 10.3934/krm.2010.3.685.
[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), 368-384. doi: 10.1137/050626478.
[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), 169-194. doi: 10.1017/S0308210500018308.
[9]	S. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations, J. Diff. Eqns., 248 (2010), 95-110. doi: 10.1016/j.jde.2009.08.016.
[10]	S. Ma, Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations, J. Math. Anal. Appl., 387(2) (2012), 1033-1043. doi: 10.1016/j.jmaa.2011.10.010.
[11]	P. L. Lions, "Mathematical Topics in Fluid Dynamics 2, Compressible Models," Oxford Science Publication, Oxford, 1998.
[12]	J. Smoller, "Shock Waves and Reaction-Diffusion Equations," $2^{nd},$ Springger-Verlag, New York, 1994.
[13]	H. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), 411-432. doi: 10.1016/j.jmaa.2004.03.064.
[14]	Z. Xin and H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations, J. Diff. Eqns., 249 (2010), 827-871. doi: 10.1016/j.jde.2010.03.011.
[15]	Z. Xin, On nonlinear stability of contact discontinuities, In "Hyperbolic Problems: Theory; Numerics, Applicaions" (Stony Brook, NY, 1994), 249-257. Word Sci. Publishing, River Edge, NJ, 1996.
[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), 621-665. doi: 0010-3640/93/050621-45.

show all references

References:

[1]	C. Duyn and L. Peletier, A class of similarity solution of the nonlinear diffusion equation, Nonlinear Analysis T.M.A., 1 (1977), 223-233. doi: 10.1016/0362-546X(77)90032-3.
[2]	J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal., 121 (1992), 235-265. doi: 10.1007/BF00410614.
[3]	D. Hoff and T. Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentripic flow with shock data, Indiana Univ. Math. J., 38 (1989), 861-915.
[4]	F. Huang, M. Li and Y. Wang, Zero dissipation limit to rarefaction wave with vacuum for 1-D compressible Navier-Stokes equations, SIAM J. Math. Anal., 44(3) (2012), 1742-1759. arXiv:1011.1991v1. doi: 10.1137/100814305.
[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), 55-77. doi: 10.1007/s00205-005-0380-7.
[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), 685-728. arXiv:1011.1990v1 doi: 10.3934/krm.2010.3.685.
[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), 368-384. doi: 10.1137/050626478.
[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), 169-194. doi: 10.1017/S0308210500018308.
[9]	S. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations, J. Diff. Eqns., 248 (2010), 95-110. doi: 10.1016/j.jde.2009.08.016.
[10]	S. Ma, Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations, J. Math. Anal. Appl., 387(2) (2012), 1033-1043. doi: 10.1016/j.jmaa.2011.10.010.
[11]	P. L. Lions, "Mathematical Topics in Fluid Dynamics 2, Compressible Models," Oxford Science Publication, Oxford, 1998.
[12]	J. Smoller, "Shock Waves and Reaction-Diffusion Equations," $2^{nd},$ Springger-Verlag, New York, 1994.
[13]	H. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), 411-432. doi: 10.1016/j.jmaa.2004.03.064.
[14]	Z. Xin and H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations, J. Diff. Eqns., 249 (2010), 827-871. doi: 10.1016/j.jde.2010.03.011.
[15]	Z. Xin, On nonlinear stability of contact discontinuities, In "Hyperbolic Problems: Theory; Numerics, Applicaions" (Stony Brook, NY, 1994), 249-257. Word Sci. Publishing, River Edge, NJ, 1996.
[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), 621-665. doi: 0010-3640/93/050621-45.

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