Hydrodynamic Limits of the Boltzmann Equation

  • Nader Masmoudi
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 135)

Abstract

From a physical point of view, we expect that a gas can be described by a fluid mechanic equation when the mean free path goes to zero. We present here some (rigorous) derivation of incompressible Fluid Mechanic equations starting from the Boltzmann equation in the limit where the free mean path (Knudsen number) goes to zero. This work can be seen as an extension of the important series of papers by C. Bardos, F. Golse and D. Levermore [1].

Keywords

Entropy Lution Incompressibility 

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References

  1. [1]
    C. Bardos, F. Golse, and C.D. Levermore. Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math., 46(5): 667–753, 1993.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    C. Bardos, F. Golse, and C.D. Levermore. Acoustic and Stokes limits for the Boltzmann equation. C. R. Acad. Sci. Paris Sero I Math., 321(3): 323–328, 1998.MathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Bardos, F. Golse, and C.D. Levermore. The acoustic limit for the Boltzmann equation. Arch. Ration. Mech. Anal., 153(3): 177–204, 2000.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    C. Bardos, F. Golse, and C.D. Levermore. Fluid dynamic limits of kinetic equations. I. Formal derivations. J. Statist. Phys., 63(1–2): 323–344, 1991.MathSciNetCrossRefGoogle Scholar
  5. [5]
    C. Bardos and S. Ukai. The classical incompressible Navier-Stokes limit of the Boltzmann equation. Math. Models Methods Appl. Sci., 1(2): 235–257, 1991.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    R.E. Caflisch. The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math., 33(5): 651–666, 1980.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    C. Cercignani. The Boltzmann equation and its applications. Springer -Verlag, New York, 1988.MATHCrossRefGoogle Scholar
  8. [8]
    C. Cercignani, R. Illner, and M. Pulvirenti. The mathematical theory of dilute gases. Springer -Verlag, New York, 1994.MATHGoogle Scholar
  9. [9]
    A. De Masi, R. Esposito, and J.L. Lebowitz. Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Comm. Pure Appl. Math., 42(8): 1189–1214, 1989.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    R.J. Diperna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math., 130(2): 321–366, 1989.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    F. Golse. From kinetic to macroscopic models. Preprint, 1998.Google Scholar
  12. [12]
    F. Golse, and C.D. Levermore. Stokes-Fourier and acoustic limits for the Boltzmann equation: convergence proofs. Comm. Pure Appl. Math., 55(3): 336–393, 2002.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    F. Golse and L. Saint-Raymond, The Navier-Stokes limit of the Boltzmann equation: convergence proof. Preprint.Google Scholar
  14. [14]
    M. Lachowicz. On the initial layer and the existence theorem for the nonlinear Boltzmann equation. Math. Methods Appl. Sci., 9(3): 342–366, 1987.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    C.D. Levermore and N. Masmoudi, From the Boltzmann Equation to an Incompressible Navier-Stokes-Fourier System, Preprint.Google Scholar
  16. [16]
    P.-L. Lions and N. Masmoudi. Une approche locale de la limite incompressible. C. R. Aead. Sci. Paris Sero I Math., 329(5): 387–392, 1999.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    P.-L. Lions and N. Masmoudi. From Boltzmann equations to incompressible fluid mechanies equation I. Arch. Ration. Mech. Anal., 158(3): 173–193, 2001.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    P.-L. Lionsand N. Masmoudi. From Boltzmann equations to incompressible fluid mechanies equation II. Arch. Ration. Mech. Anal., 158(3): 195–211, 2001.CrossRefGoogle Scholar
  19. [19]
    S. Ukai and K. Asano. The Euler limit and initial layer of the nonlinear Boltzmann equation. Hokkaido Math. J., 12(3, Part 1): 311–332, 1983.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Nader Masmoudi
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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