# 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 and 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), 373-392. [2] L. Boltzmann, (translated by Stephen G. Brush), "Lectures on Gas Theory," Dover Publications, Inc. New York, 1964. [3] R. E. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 491-508. [4] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, Berlin, 1994. [5] S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," 3rd edition, Cambridge University Press, 1990. [6] C. T. Duyn and L. A. Peletier, A class of similarity solution of the nonlinear diffusion equation, Nonlinear Analysis, T.M.A., 1 (1977), 223-233. [7] R. Esposito and M. Pulvirenti, From particle to fluids, in "Handbook of Mathematical Fluid Dynamics," Vol. III, North-Holland, Amsterdam, (2004), 1-82. [8] F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation, Arch. Ration. Mech. Anal., 103 (1986), 81-96. [9] J. Goodman and Z. P. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal., 121 (1992), 235-265. [10] H. Grad, "Asymptotic Theory of the Boltzmann Equation II," in "Rarefied Gas Dynamics" (J. A. Laurmann, ed.), Vol. 1, Academic Press, New York, (1963), 26-59. [11] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. [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), 861-915. [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), 89-116. [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), 55-77. [15] F. M. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phy., 295 (2010), 293-326. [16] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297. [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), 368-384. [18] M. Lachowicz, On the initial layer and existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl.Sci., 9 (1987), 342-366. [19] T. Liu, T. Yang, and S. H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192. [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), 333-371. [21] T. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246 (2004), 133-179. [22] S. X. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations, J. Diff. Eqs., 248 (2010), 95-110. [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), 1-13. [24] T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Commun. Math. Phys., 61 (1978), 119-148. [25] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994. [26] S. Ukai and K. Asano, The Euler limit and the initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 303-324. [27] S. Ukai, T. Yang and H. J. Zhao, Global solutions to the Boltzmann equation with external forces, Analysis and Applications, 3 (2005), 157-193. [28] H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), 411-432. [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), 727-748. [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), 621-665. [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), 827-871. [32] S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations, Commun. Pure Appl. Math, 58 (2005), 409-443.

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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), 373-392. [2] L. Boltzmann, (translated by Stephen G. Brush), "Lectures on Gas Theory," Dover Publications, Inc. New York, 1964. [3] R. E. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 491-508. [4] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, Berlin, 1994. [5] S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," 3rd edition, Cambridge University Press, 1990. [6] C. T. Duyn and L. A. Peletier, A class of similarity solution of the nonlinear diffusion equation, Nonlinear Analysis, T.M.A., 1 (1977), 223-233. [7] R. Esposito and M. Pulvirenti, From particle to fluids, in "Handbook of Mathematical Fluid Dynamics," Vol. III, North-Holland, Amsterdam, (2004), 1-82. [8] F. Golse, B. Perthame and C. Sulem, On a boundary layer problem for the nonlinear Boltzmann equation, Arch. Ration. Mech. Anal., 103 (1986), 81-96. [9] J. Goodman and Z. P. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rational Mech. Anal., 121 (1992), 235-265. [10] H. Grad, "Asymptotic Theory of the Boltzmann Equation II," in "Rarefied Gas Dynamics" (J. A. Laurmann, ed.), Vol. 1, Academic Press, New York, (1963), 26-59. [11] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. [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), 861-915. [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), 89-116. [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), 55-77. [15] F. M. Huang, Y. Wang and T. Yang, Hydrodynamic limit of the Boltzmann equation with contact discontinuities, Comm. Math. Phy., 295 (2010), 293-326. [16] F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297. [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), 368-384. [18] M. Lachowicz, On the initial layer and existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl.Sci., 9 (1987), 342-366. [19] T. Liu, T. Yang, and S. H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192. [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), 333-371. [21] T. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246 (2004), 133-179. [22] S. X. Ma, Zero dissipation limit to strong contact discontinuity for the 1-D compressible Navier-Stokes equations, J. Diff. Eqs., 248 (2010), 95-110. [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), 1-13. [24] T. Nishida, Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation, Commun. Math. Phys., 61 (1978), 119-148. [25] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994. [26] S. Ukai and K. Asano, The Euler limit and the initial layer of the nonlinear Boltzmann equation, Hokkaido Math. J., 12 (1983), 303-324. [27] S. Ukai, T. Yang and H. J. Zhao, Global solutions to the Boltzmann equation with external forces, Analysis and Applications, 3 (2005), 157-193. [28] H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), 411-432. [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), 727-748. [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), 621-665. [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), 827-871. [32] S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations, Commun. Pure Appl. Math, 58 (2005), 409-443.
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