Archive for Rational Mechanics and Analysis

, Volume 143, Issue 3, pp 273–307 | Cite as

On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations

  • Cédric Villani


This paper deals with the spatially homogeneous Boltzmann equation when grazing collisions are involved.We study in a unified setting the Boltzmann equation without cut-off, the Fokker-Planck-Landau equation, and the asymptotics of grazing collisions for a very broad class of potentials; in particular, we are able to derive rigorously the Landau equation for the Coulomb potential. In order to do so, we introduce a new definition of weak solutions, based on entropy production.


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  1. 1.
    L. Arkeryd: On the Boltzmann equation. Arch. Rational Mech. Anal., 45: 1–34, 1972.ADSMATHMathSciNetGoogle Scholar
  2. 2.
    L. Arkeryd: Intermolecular forces of infinite range and the Boltzmann equation. Arch. Rational Mech. Anal., 77: 11–21, 1981.ADSCrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    L. Arkeryd: Asymptotic behaviour of the Boltzmann equation with infinite range forces. Comm. Math. Phys., 86: 475–484, 1982.ADSCrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    A. A. Arsen'ev & O.E. Buryak: On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation. Math. USSR Sbornik, 69: 465–478, 1991.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    R. Balescu: Equilibrium and nonequilibrium statistical mechanics. Wiley, 1975.MATHGoogle Scholar
  6. 6.
    N. N. Bogoliubov: Problems of dynamical theory in statistical physics, in Studies in Statistical Mechanics, J. de Boer & G.E. Uhlenbeck, eds. Interscience, New York, 1962.Google Scholar
  7. 7.
    C. Buet, S. Cordier, P. Degond & M. Lemou: Fast algorithms for numerical conservative and entropy approximations of the Fokker-Planck equation in 3D velocity space. J. Comput. Phys., 133: 310–322, 1997.ADSCrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    R. Caflisch: The Boltzmann equation with a soft potential. Commun. Math. Phys., 74: 71–109, 1980.ADSCrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    T. Carleman: Problèmes mathématiques dans la théorie cinétiquedes gaz. Almqvist & Wiksell, 1957.Google Scholar
  10. 10.
    C. Cercignani: The Boltzmann equation and its applications. Springer, New York, 1988.CrossRefMATHGoogle Scholar
  11. 11.
    C. Cercignani, R. Illner, & M. Pulvirenti: The mathematical theory of dilute gases. Springer, New York, 1994.CrossRefMATHGoogle Scholar
  12. 12.
    P. Degond & B. Lucquin-Desreux: The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Mod. Meth. in Appl. Sci., 2: 167–182, 1992.MATHMathSciNetGoogle Scholar
  13. 13.
    L. Desvillettes: On asymptotics of the Boltzmann equation when the collisions become grazing. Transp. Theory Stat. Phys., 21: 259–276, 1992.ADSCrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    L. Desvillettes: Some applications of the method of moments for the homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 123: 387–395, 1993.ADSCrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    L. Desvillettes: About the regularizing properties of the non-cut-off Kac equation. Comm. Math. Phys., 168: 417–440, 1995.ADSCrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    L. Desvillettes & C. Villani: On the spatially homogeneous Landau equation with hard potentials. Work in preparation.Google Scholar
  17. 17.
    R. J. Di Perna & P. L. Lions: On the Fokker-Planck-Boltzmann equation. Commun. Math. Phys., 120: 1–23, 1988.ADSGoogle Scholar
  18. 18.
    R. J. Di Perna & P. L. Lions: On the Cauchy problem for the Boltzmann equation: global existence and weak stability. Ann. Math., 130: 312–366, 1989.Google Scholar
  19. 19.
    T. Elmroth: Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rational Mech. Anal., 82: 1–12, 1983.ADSCrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    T. Goudon: Generalized invariant sets for the Boltzmann equation. Math. Mod. Meth. Appl. Sci., 7: 457–476, 1997.CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    T. Goudon: On the Boltzmann equation and its relations to the Landau-Fokker-Planck equation: influence of grazing collisions. C. R. Acad. Sci. Paris, 324, Série I: 265–270, 1997.ADSCrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    T. Gustafsson: Global Lp-properties for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal., 103: 1–38, 1988.ADSCrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    E. M. Lifchitz & L.P. Pitaevskii: Physical Kinetics — Course in theoretical physics, volume 10. Pergamon, Oxford, 1981.Google Scholar
  24. 24.
    P. L. Lions: On Boltzmann and Landau equations. Phil. Trans. R. Soc. Lond., A 346: 191–204, 1994.ADSCrossRefMATHGoogle Scholar
  25. 25.
    J. C. Maxwell: On the dynamical theory of gases. Phil. Trans. R. Soc. Lond., 157: 49–88, 1866.CrossRefGoogle Scholar
  26. 26.
    S. Mischler & B. Wennberg: On the spatially homogeneous Boltzmann equation. To appear in Ann. IHP.Google Scholar
  27. 27.
    A. Ja. Povzner: The Boltzmann equation in the kinetic theory of gases. Amer. Math. Soc. Trans., 47 (Ser. 2) 193–214, 1965.MATHGoogle Scholar
  28. 28.
    A. Proutière: New results of regularization for weak solutions of Boltzmann equation, preprint, 1996.Google Scholar
  29. 29.
    I. P. Shkarofsky, T. W. Johnston & M.P. Bachynski: The particle kinetics of plasmas. Addison-Wesley, Reading, 1966.Google Scholar
  30. 30.
    C. Truesdell & R.G. Muncaster: Fundamentals of Maxwell’s kinetic theory of a simple monatomic gas. Academic Press, New York, 1980.Google Scholar
  31. 31.
    S. Ukai & K. Asano: On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Math. Sci., 18: 477–519, 1982.CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    C. Villani: On the Landau equation: weak stability, global existence. Adv. Diff. Eq., 1: 793–816, 1996.MATHMathSciNetGoogle Scholar
  33. 33.
    C. Villani: On the spatially homogeneous Landau equation for Maxwellian molecules, to appear in Math. Mod. Meth. Appl. Sci. Google Scholar
  34. 34.
    B. Wennberg: On moments and uniqueness for solutions to the space homogeneous Boltzmann equation. Transp. Theory. Stat. Phys., 24: 533–539, 1994.ADSCrossRefMathSciNetGoogle Scholar
  35. 35.
    H. E. Wilhelm: Momentum and energy exchange between beams of particles interacting by Yukawa-type potentials. Phys. Rev., 187: 382–392, 1969.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Cédric Villani
    • 1
  1. 1.Ecole Normale SupérieureParis Cedex 05France

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