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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 & 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.  doi: 10.1016/0362-546X(77)90032-3.  Google Scholar

[2]

J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal.,, \textbf{121} (1992), 121 (1992), 235.  doi: 10.1007/BF00410614.  Google Scholar

[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.   Google Scholar

[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 (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.  Google Scholar

[9]

S. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations,, J. Diff. Eqns., 248 (2010), 95.  doi: 10.1016/j.jde.2009.08.016.  Google Scholar

[10]

S. Ma, Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations,, J. Math. Anal. Appl., 387 (2012), 1033.  doi: 10.1016/j.jmaa.2011.10.010.  Google Scholar

[11]

P. L. Lions, "Mathematical Topics in Fluid Dynamics 2, Compressible Models,", Oxford Science Publication, (1998).   Google Scholar

[12]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", $2^{nd}, (1994).   Google Scholar

[13]

H. Wang, Viscous limits for piecewise smooth solutions of the p-system,, J. Math. Anal. Appl., 299 (2004), 411.  doi: 10.1016/j.jmaa.2004.03.064.  Google Scholar

[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.  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

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.  doi: 10.1016/0362-546X(77)90032-3.  Google Scholar

[2]

J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Ration. Mech. Anal.,, \textbf{121} (1992), 121 (1992), 235.  doi: 10.1007/BF00410614.  Google Scholar

[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.   Google Scholar

[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 (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.  Google Scholar

[9]

S. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations,, J. Diff. Eqns., 248 (2010), 95.  doi: 10.1016/j.jde.2009.08.016.  Google Scholar

[10]

S. Ma, Viscous limit to contact discontinuity for the 1-D compressible Navier-Stokes equations,, J. Math. Anal. Appl., 387 (2012), 1033.  doi: 10.1016/j.jmaa.2011.10.010.  Google Scholar

[11]

P. L. Lions, "Mathematical Topics in Fluid Dynamics 2, Compressible Models,", Oxford Science Publication, (1998).   Google Scholar

[12]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", $2^{nd}, (1994).   Google Scholar

[13]

H. Wang, Viscous limits for piecewise smooth solutions of the p-system,, J. Math. Anal. Appl., 299 (2004), 411.  doi: 10.1016/j.jmaa.2004.03.064.  Google Scholar

[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.  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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