On boltzmann equations and fokker—planck asymptotics: Influence of grazing collisions

  • T. Goudon


In this paper, we are interested in the influence of grazing collisions, with deflection angle near π/2, in the space-homogeneous Boltzmann equation. We consider collision kernels given by inverse-sth-power force laws, and we deal with general initial data with bounded mass, energy, and entropy. First, once a suitable weak formulation is defined, we prove the existence of solutions of the spatially homogeneous Boltzmann equation, without angular cutoff assumption on the collision kernel, fors ≥ 7/3. Next, the convergence of these solutions to solutions of the Landau-Fokker-Planck equation is studied when the collision kernel concentrates around the value π/2. For very soft interactions, 2<s<7/3, the existence of weak solutions is discussed concerning the Boltzmann equation perturbed by a diffusion term

Key Words

Kinetic equation homogeneous Boltzmann equation Landau-Fokker-Planck equation grazing collisions 


  1. 1.
    L. Arkeryd, On the Boltzmann equation,Arch. Rat. Mech. Anal. 45:1–34 (1972).MATHMathSciNetGoogle Scholar
  2. 2.
    L. Arkeryd, Intermolecular forces of infinite range and the Boltzmann equation,Arch. Rat. Mech. Anal. 77:11–21 (1981).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. Arsenev and O. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation,Math. Sbornyk 181:4, 465–477 (1991).Google Scholar
  4. 4.
    A. Bers and J. L. Delcroix,Physique des plasmas (InterEdition/CNRS, 1994).Google Scholar
  5. 5.
    H. Brezis,Analyse fonctionnelle (Masson, 1993).Google Scholar
  6. 6.
    C. Cercignani,The Boltzmann equation and its applications (Springer-Verlag, 1988).Google Scholar
  7. 7.
    C. Cercignani, R. Illner, and M. Pulvirenti,The mathematical theory of dilute gases (Springer-Verlag, 1994).Google Scholar
  8. 8.
    F. Chvala and R. Pettersson, Weak solutions of the linear Boltzmann equation with very soft interactions,J. Math. Anal. and Appl. 191:360–379 (1995).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    F. Chvala, T. Gustafsson, and R. Pettersson, On solutions to the linear Boltzmann equation with external electromagnetic force,SIAM J. Math. Anal. 24:3, 583–602 (1993).CrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann operator in the Coulomb case,Math. Models and Meth. in the Appl. Sci. 2:2, 167–182 (1992).MathSciNetGoogle Scholar
  11. 11.
    L. Desvillettes, On the asymptotics of the Boltzmann equation when the collisions become grazing,Transp. Theory in Stat. Phys. 21:259–276 (1992).MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations,Arch. Rat. Mech. Anal. 123:387–404 (1993).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    R. Di Perna and P. L. Lions, On the Cauchy problem for Boltzmann equation: Global existence and weak stability,Ann. Math. 130 :321–366 (1989).CrossRefGoogle Scholar
  14. 14.
    R. Di Perna and P.-L. Lions, On the Fokker-Planck-Boltzmann equation,Comm. Math. Phys. 120:1–23 (1988).CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    T. Elmroth, Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range,Arch. Rat. Mech. Anal. 82:1–12 (1983).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    T. Goudon, Sur l’équation de Boltzmann homogène et sa relation avec l’équation de Landau-Fokker-Planck: Influence des collisions rasantes,CRAS 324 :265–270 (1997).MATHMathSciNetGoogle Scholar
  17. 17.
    T. Goudon,Sur quelques questions relatives à la théorie cinétique des gaz et à l’équation de Boltzmann, Thèse Université Bordeaux 1 (1997).Google Scholar
  18. 18.
    H. Grad, Asymptotic theory of the Boltzmann equation inRarefied Gas Dynamic, Third Symposium, Paris, Laurmann Ed., pp. 26–59 (Ac. Press 1963).Google Scholar
  19. 19.
    K. Hamdache,Sur l’existence globale et le comportement asymptotique de quelques solutions de l’equation de Boltzmann, Thèse Université Paris 6 (1986).Google Scholar
  20. 20.
    K. Hamdache, Estimations uniformes des solutions de l’équation de Boltzmann par les méthodes de viscosité artificielle et de diffusion de Fokker-Planck,CRAS 302:187–190 (1986).MATHMathSciNetGoogle Scholar
  21. 21.
    L. Hormander,The analysis of linear pde, Vol. 1 (Springer, 1983).Google Scholar
  22. 22.
    N. A. Krall and A. W. Trivelpiece,Principles of plasma physics (Mc Graw-Hill, 1964).Google Scholar
  23. 23.
    E. Lifshitz and L. Pitaevski,Cinétique Physique. Coll. “Physique Théorique,” L. Landau-E. Lifshitz (Mir, 1990).Google Scholar
  24. 24.
    P. L. Lions, Compactness in Boltzmann equation via Fourier integral operators and applications. Part I, II and III,J. Math. Kyoto Univ. 34:2, (39) 391–461 (1994), andJ. Math. Kyoto Univ. 34 :3, 539–584 (1994).Google Scholar
  25. 25.
    R. Pettersson, Existence theorems for the linear space homogeneous transport equation,IMA J. Appl. Math. 30:81–105 (1983).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    R. Pettersson, On solutions and higher moments for the linear Boltzmann equation with infinite-range forces,IMA J. Appl. Math. 38:151–166 (1987).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    R. Pettersson, On solutions to the linear Boltzmann equation with general boundary conditions and infinite-range forces,J. Stat. Phys. 59:1/2, 403–440 (1990).MATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    M. Reed and B. Siimon,Methods of modern mathematical physics., Vol. 2 (Ac. Press, 1970).Google Scholar
  29. 29.
    S. Semmes, A primer on Hardy spaces,Comm. Part. Diff. Eqt. 19:277–319 (1994).MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    J. Simon, Compact sets in Lp(0,T;B), Ann. Mat. Pura Appl. IV:146, 65–96 (1987).Google Scholar
  31. 31.
    S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,Japan J. Appl. Math. 1:141–156 (1984).MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    C. Villani,On a new class of weak solutions to the Boltzmann and Landau equations. Personal communication (1997).Google Scholar
  33. 33.
    B. Wennberg, On moments and uniqueness for solutions to the space homogeneous Boltzmann equation,Transp. Th. Stat. Phys. 24:4, 533–539 (1994).MathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • T. Goudon
    • 1
  1. 1.Mathématiques Appliquées de BordeauxCNRS-Université Bordeaux ITalence CedexFrance

Personalised recommendations