# American Institute of Mathematical Sciences

December  2010, 3(4): 685-728. doi: 10.3934/krm.2010.3.685

## Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity

 1 Institute of Applied Mathematics, AMSS and Hua Loo-Keng Key Laboratory of Mathematics, Academia Sinica, Beijing 100190 2 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  August 2010 Revised  October 2010 Published  October 2010

Fluid dynamic limit to compressible Euler equations from compressible Navier-Stokes equations and Boltzmann equation has been an active topic with limited success so far. In this paper, we consider the case when the solution of the Euler equations is a Riemann solution consisting two rarefaction waves and a contact discontinuity and prove this limit for both Navier-Stokes equations and the Boltzmann equation when the viscosity, heat conductivity coefficients and the Knudsen number tend to zero respectively. In addition, the uniform convergence rates in terms of the above physical parameters are also obtained. It is noted that this is the first rigorous proof of this limit for a Riemann solution with superposition of three waves even though the fluid dynamic limit for a single wave has been proved.
Citation: Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic & Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
##### References:
 [1] F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation,, Arch. Rat. Mech. Anal., 54 (1974), 373.   Google Scholar [2] L. Boltzmann, (translated by Stephen G. Brush), "Lectures on Gas Theory,", Dover Publications, (1964).   Google Scholar [3] R. E. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation,, Comm. Pure Appl. Math., 33 (1980), 491.   Google Scholar [4] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Springer-Verlag, (1994).   Google Scholar [5] S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", 3rd edition, (1990).   Google Scholar [6] C. T. Duyn and L. A. Peletier, A class of similarity solution of the nonlinear diffusion equation,, Nonlinear Analysis, 1 (1977), 223.   Google Scholar [7] R. Esposito and M. Pulvirenti, From particle to fluids,, in, III (2004), 1.   Google Scholar [8] F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation,, Arch. Ration. Mech. Anal., 103 (1986), 81.   Google Scholar [9] J. Goodman and Z. P. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws,, Arch. Rational Mech. Anal., 121 (1992), 235.   Google Scholar [10] H. Grad, "Asymptotic Theory of the Boltzmann Equation II,", in, 1 (1963), 26.   Google Scholar [11] Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.   Google Scholar [12] D. Hoff and T. P. Liu, The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data,, India. Univ. Math. J., 36 (1989), 861.   Google Scholar [13] F. M. Huang, J. Li and A. Matsumura, Stability of the combination of the viscous contact wave and the rarefaction wave to the compressible Navier-Stokes equations,, Arch. Ration. Mech. Anal., 197 (2010), 89.   Google Scholar [14] F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations,, Arch. Rat. Mech. Anal., 179 (2006), 55.   Google Scholar [15] F. M. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities,, Comm. Math. Phy., 295 (2010), 293.   Google Scholar [16] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion,, Adv. Math., 219 (2008), 1246.   Google Scholar [17] S. Jiang, G. X. Ni and W. J. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equiations of one-dimensional compressible heat-conducting fluids,, SIAM J. Math. Anal., 38 (2006), 368.   Google Scholar [18] M. Lachowicz, On the initial layer and existence theorem for the nonlinear Boltzmann equation,, Math. Methods Appl.Sci., 9 (1987), 342.   Google Scholar [19] T. Liu, T. Yang, and S. H. Yu, Energy method for the Boltzmann equation,, Physica D, 188 (2004), 178.   Google Scholar [20] T. Liu, T. Yang, S. H. Yu and H. J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation,, Arch. Rat. Mech. Anal., 181 (2006), 333.   Google Scholar [21] T. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles,, Commun. Math. Phys., 246 (2004), 133.   Google Scholar [22] S. X. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations,, J. Diff. Eqs., 248 (2010), 95.   Google Scholar [23] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas,, Japan J. Appl. Math., 3 (1986), 1.   Google Scholar [24] T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation,, Commun. Math. Phys., 61 (1978), 119.   Google Scholar [25] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, (1994).   Google Scholar [26] S. Ukai and K. Asano, The Euler limit and the initial layer of the nonlinear Boltzmann equation,, Hokkaido Math. J., 12 (1983), 303.   Google Scholar [27] S. Ukai, T. Yang and H. J. Zhao, Global solutions to the Boltzmann equation with external forces,, Analysis and Applications, 3 (2005), 157.   Google Scholar [28] H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system,, J. Math. Anal. Appl., 299 (2004), 411.   Google Scholar [29] Y. Wang, Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock,, Acta Mathematica Scientia Ser. B, 28 (2008), 727.   Google Scholar [30] Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimentional Navier-Stokes equations of compressible isentropic gases,, Commun. Pure Appl. Math, XLVI (1993), 621.   Google Scholar [31] Z. P. Xin and H. H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations,, J. Diff. Eqs., 249 (2010), 827.   Google Scholar [32] S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations,, Commun. Pure Appl. Math, 58 (2005), 409.   Google Scholar

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##### References:
 [1] F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation,, Arch. Rat. Mech. Anal., 54 (1974), 373.   Google Scholar [2] L. Boltzmann, (translated by Stephen G. Brush), "Lectures on Gas Theory,", Dover Publications, (1964).   Google Scholar [3] R. E. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation,, Comm. Pure Appl. Math., 33 (1980), 491.   Google Scholar [4] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Springer-Verlag, (1994).   Google Scholar [5] S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,", 3rd edition, (1990).   Google Scholar [6] C. T. Duyn and L. A. Peletier, A class of similarity solution of the nonlinear diffusion equation,, Nonlinear Analysis, 1 (1977), 223.   Google Scholar [7] R. Esposito and M. Pulvirenti, From particle to fluids,, in, III (2004), 1.   Google Scholar [8] F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation,, Arch. Ration. Mech. Anal., 103 (1986), 81.   Google Scholar [9] J. Goodman and Z. P. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws,, Arch. Rational Mech. Anal., 121 (1992), 235.   Google Scholar [10] H. Grad, "Asymptotic Theory of the Boltzmann Equation II,", in, 1 (1963), 26.   Google Scholar [11] Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081.   Google Scholar [12] D. Hoff and T. P. Liu, The inviscid limit for the Navier-Stokes equations of compressible isentropic flow with shock data,, India. Univ. Math. J., 36 (1989), 861.   Google Scholar [13] F. M. Huang, J. Li and A. Matsumura, Stability of the combination of the viscous contact wave and the rarefaction wave to the compressible Navier-Stokes equations,, Arch. Ration. Mech. Anal., 197 (2010), 89.   Google Scholar [14] F. M. Huang, A. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations,, Arch. Rat. Mech. Anal., 179 (2006), 55.   Google Scholar [15] F. M. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities,, Comm. Math. Phy., 295 (2010), 293.   Google Scholar [16] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion,, Adv. Math., 219 (2008), 1246.   Google Scholar [17] S. Jiang, G. X. Ni and W. J. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes equiations of one-dimensional compressible heat-conducting fluids,, SIAM J. Math. Anal., 38 (2006), 368.   Google Scholar [18] M. Lachowicz, On the initial layer and existence theorem for the nonlinear Boltzmann equation,, Math. Methods Appl.Sci., 9 (1987), 342.   Google Scholar [19] T. Liu, T. Yang, and S. H. Yu, Energy method for the Boltzmann equation,, Physica D, 188 (2004), 178.   Google Scholar [20] T. Liu, T. Yang, S. H. Yu and H. J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation,, Arch. Rat. Mech. Anal., 181 (2006), 333.   Google Scholar [21] T. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles,, Commun. Math. Phys., 246 (2004), 133.   Google Scholar [22] S. X. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations,, J. Diff. Eqs., 248 (2010), 95.   Google Scholar [23] A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction wave of the solutions of a one-dimensional model system for compressible viscous gas,, Japan J. Appl. Math., 3 (1986), 1.   Google Scholar [24] T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation,, Commun. Math. Phys., 61 (1978), 119.   Google Scholar [25] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, (1994).   Google Scholar [26] S. Ukai and K. Asano, The Euler limit and the initial layer of the nonlinear Boltzmann equation,, Hokkaido Math. J., 12 (1983), 303.   Google Scholar [27] S. Ukai, T. Yang and H. J. Zhao, Global solutions to the Boltzmann equation with external forces,, Analysis and Applications, 3 (2005), 157.   Google Scholar [28] H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system,, J. Math. Anal. Appl., 299 (2004), 411.   Google Scholar [29] Y. Wang, Zero dissipation limit of the compressible heat-conducting Navier-Stokes equations in the presence of the shock,, Acta Mathematica Scientia Ser. B, 28 (2008), 727.   Google Scholar [30] Z. P. Xin, Zero dissipation limit to rarefaction waves for the one-dimentional Navier-Stokes equations of compressible isentropic gases,, Commun. Pure Appl. Math, XLVI (1993), 621.   Google Scholar [31] Z. P. Xin and H. H. Zeng, Convergence to the rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations,, J. Diff. Eqs., 249 (2010), 827.   Google Scholar [32] S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations,, Commun. Pure Appl. Math, 58 (2005), 409.   Google Scholar
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