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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

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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