American Institute of Mathematical Sciences

Advanced
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

Feimin Huang 1, , Yi Wang 1, and  Tong Yang 2,

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.
Keywords: Compressible Navier-Stokes equations, fluid dynamic limit., Boltzmann equation, contact discontinuity, rarefaction wave.
Mathematics Subject Classification: Primary: 35Q30, 35Q20, 76N15, 76P05; Secondary: 35L65, 82B40, 82C4.
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), 373-392.  Google Scholar

[2]

L. Boltzmann, (translated by Stephen G. Brush), "Lectures on Gas Theory," Dover Publications, Inc. New York, 1964.  Google Scholar

[3]

R. E. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 491-508.  Google Scholar

[4]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, Berlin, 1994.  Google Scholar

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," 3rd edition, Cambridge University Press, 1990.  Google Scholar

[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.  Google Scholar

[7]

R. Esposito and M. Pulvirenti, From particle to fluids, in "Handbook of Mathematical Fluid Dynamics," Vol. III, North-Holland, Amsterdam, (2004), 1-82.  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), 81-96.  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), 235-265.  Google Scholar

[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.  Google Scholar

[11]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.  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), 861-915.  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), 89-116.  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), 55-77.  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), 293-326.  Google Scholar

[16]

F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297.  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), 368-384.  Google Scholar

[18]

M. Lachowicz, On the initial layer and existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl.Sci., 9 (1987), 342-366.  Google Scholar

[19]

T. Liu, T. Yang, and S. H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192.  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), 333-371.  Google Scholar

[21]

T. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246 (2004), 133-179.  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), 95-110.  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), 1-13.  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), 119-148.  Google Scholar

[25]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 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), 303-324.  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), 157-193.  Google Scholar

[28]

H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), 411-432.  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), 727-748.  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), 621-665.  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), 827-871. Google Scholar

[32]

S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations, Commun. Pure Appl. Math, 58 (2005), 409-443.  Google Scholar

show all references

References:
[1]

F. V. Atkinson and L. A. Peletier, Similarity solutions of the nonlinear diffusion equation, Arch. Rat. Mech. Anal., 54 (1974), 373-392.  Google Scholar

[2]

L. Boltzmann, (translated by Stephen G. Brush), "Lectures on Gas Theory," Dover Publications, Inc. New York, 1964.  Google Scholar

[3]

R. E. Caflish, The fluid dynamical limit of the nonlinear Boltzmann equation, Comm. Pure Appl. Math., 33 (1980), 491-508.  Google Scholar

[4]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Springer-Verlag, Berlin, 1994.  Google Scholar

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases," 3rd edition, Cambridge University Press, 1990.  Google Scholar

[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.  Google Scholar

[7]

R. Esposito and M. Pulvirenti, From particle to fluids, in "Handbook of Mathematical Fluid Dynamics," Vol. III, North-Holland, Amsterdam, (2004), 1-82.  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), 81-96.  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), 235-265.  Google Scholar

[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.  Google Scholar

[11]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.  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), 861-915.  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), 89-116.  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), 55-77.  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), 293-326.  Google Scholar

[16]

F. M. Huang, Z. P. Xin and T. Yang, Contact discontinuities with general perturbation for gas motion, Adv. Math., 219 (2008), 1246-1297.  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), 368-384.  Google Scholar

[18]

M. Lachowicz, On the initial layer and existence theorem for the nonlinear Boltzmann equation, Math. Methods Appl.Sci., 9 (1987), 342-366.  Google Scholar

[19]

T. Liu, T. Yang, and S. H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192.  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), 333-371.  Google Scholar

[21]

T. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Commun. Math. Phys., 246 (2004), 133-179.  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), 95-110.  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), 1-13.  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), 119-148.  Google Scholar

[25]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 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), 303-324.  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), 157-193.  Google Scholar

[28]

H. Y. Wang, Viscous limits for piecewise smooth solutions of the p-system, J. Math. Anal. Appl., 299 (2004), 411-432.  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), 727-748.  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), 621-665.  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), 827-871. Google Scholar

[32]

S. H. Yu, Hydrodynamic limits with shock waves of the Boltzmann equations, Commun. Pure Appl. Math, 58 (2005), 409-443.  Google Scholar

[1]

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
[2]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[3]

Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021
[4]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985
[5]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[6]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[7]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
[8]

Li Fang, Zhenhua Guo. Zero dissipation limit to rarefaction wave with vacuum for a one-dimensional compressible non-Newtonian fluid. Communications on Pure & Applied Analysis, 2017, 16 (1) : 209-242. doi: 10.3934/cpaa.2017010
[9]

Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409
[10]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[11]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[12]

Daoyuan Fang, Ting Zhang. Compressible Navier-Stokes equations with vacuum state in one dimension. Communications on Pure & Applied Analysis, 2004, 3 (4) : 675-694. doi: 10.3934/cpaa.2004.3.675
[13]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595
[14]

Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609
[15]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[16]

Dong Li, Xinwei Yu. On some Liouville type theorems for the compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4719-4733. doi: 10.3934/dcds.2014.34.4719
[17]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[18]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001
[19]

Linglong Du, Haitao Wang. Pointwise wave behavior of the Navier-Stokes equations in half space. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055
[20]

Fabio Ramos, Edriss S. Titi. Invariant measures for the $3$D Navier-Stokes-Voigt equations and their Navier-Stokes limit. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 375-403. doi: 10.3934/dcds.2010.28.375

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (108)
  • HTML views (0)
  • Cited by (35)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy