Journal of Statistical Physics

, Volume 104, Issue 1–2, pp 327–358 | Cite as

Some Solutions of the Boltzmann Equation Without Angular Cutoff

  • Radjesvarane Alexandre


We show the existence of local or global in time solutions for the non-homogeneous Boltzmann equation. This is done under the assumptions that initial data are smaller than a suitable Maxwellian and that collisional cross-sections do not satisfy Grad's angular cutoff. Partial regularity in space-velocity of the solutions constructed herein is also proved.

Boltzmann singular cross-sections 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ale1]
    R. Alexandre, Une définition des solutions renormalisées pour l'équation de Boltzmann, Note C.R.A.S Paris, t. 328, Série I, pp. 987–991 (1999).Google Scholar
  2. [Ale2]
    R. Alexandre, work in preparation.Google Scholar
  3. [Ale3]
    R. Alexandre, Around 3D Boltzmann operator without cutoff. A New formulation. M 2AN, Vol. 34, No.3, pp. 575–590 (2000).Google Scholar
  4. [Ale4]
    R. Alexandre, From Boltzmann to Landau, SIAM J. Appl. Maths. (1999), submitted.Google Scholar
  5. [ADVW]
    R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long range interactions, Arch. Rat. Mech. Anal. 152(4):327–355 (2000).Google Scholar
  6. [AlVi]
    R. Alexandre and C. Villani, On the Boltzmann equation with long range interactions and the Landau approximation in plasma physics. Preprint DMI-ENS Paris (1999). First part submitted to C.P.A.M.Google Scholar
  7. [ArBe]
    L. Arlotti and N. Bellomo, Lectures Notes on the Math. Theory of the Boltzmann Equation, N. Bellomo, ed., Vol. 33 (World Sc., 1995).Google Scholar
  8. [Bal]
    R. Balescu, Statistical Mechanics of Charged Particles (Wiley Interscience, N.Y., 1963).Google Scholar
  9. [Cer]
    C. Cercignani, Mathematical Methods in Kinetic Theory, 2nd ed. (Plenum, 1990).Google Scholar
  10. [CIP]
    C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, series in Appl. Sc., Vol. 106 (Springer Verlag, New York, 1994).Google Scholar
  11. [Des1]
    L. Desvillettes, Regularisation for the non cutoff 2D radially symmetric Boltzmann equation with a velocity dependent cross section, Transp. Theory and Stat. Phys. 25(3–5):383–394 (1996).Google Scholar
  12. [Des2]
    L. Desvillettes, Regularisation properties of the 2D non radially symmetric non cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Transp. Theory and Stat. Phys. 26(3):341–357 (1997).Google Scholar
  13. [DeGo]
    L. Desvillettes and F. Golse, On a model Boltzmann equation without angular cutoff, preprint, Univ. Orléans 98-01 (1998).Google Scholar
  14. [DeVi]
    L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials, preprint DMI-ENS Paris (1998).Google Scholar
  15. [DiLi1]
    R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equation; Global existence and weak stability, Ann. Maths. 130:321–366 (1989).Google Scholar
  16. [DiLi2]
    R. J. DiPerna and P. L. Lions, On the Fokker–Planck Boltzmann equation, Com. Math. Phys. 120:1–23 (1988).Google Scholar
  17. [Gou]
    T. Goudon, Generalized invariants sets for the Boltzmann equation, M 3AS 7(4):457–476 (1997).Google Scholar
  18. [Ham]
    K. Hamdache, Sur l'existence globale de solutions de l'équation de Boltzmann. Thèse d'Etat. Univ. Paris 6, Paris (1986).Google Scholar
  19. [IlSh]
    R. Illner and M. Shinbrot, The Boltzmann equation. Global existence for a rare gas in an infinite vacuum, Com. Math. Phys. 95:217–226 (1984).Google Scholar
  20. [Lio1]
    P. L. Lions, Compactness in Boltzmann's equation, via FIO and applications, J. Math. Kyoto Univ. Part I 34:391–427; Part II 34:429–461; Part III 34:539–584 (1994).Google Scholar
  21. [Lio2]
    P. L. Lions, On Boltzmann and Landau equation, Phil. Trans. Roy. Soc. London A 346:191–204 (1994).Google Scholar
  22. [Lio3]
    P. L. Lions, Régularité et compacité pour des noyaux de collisions de Boltzmann sans troncature angulaire, Note C.R.A.S Paris, t. {nu326 , Série I, pp. 37–41 (1998).Google Scholar
  23. [Mar1]
    J. Marschall, Pdo with non regular symbols of the class S{in \(S_{\varrho ,\delta }^m ,\), Com. Part. Diff. Equ. 12 (8): 921–965 (1987).Google Scholar
  24. [Mar2]
    J. Marschall, Pdo with coefficients in Sobolev spaces, Trans. AMS 307 (1):335–361 (1988).Google Scholar
  25. [Per]
    B. Perthame, Higher moments for kinetic equations for Vlasov–Poisson and Fokker–Planck cases, M2AS 13:441–452 (1990).Google Scholar
  26. [Tay1]
    M. E. Taylor, Pseudo-Differential Operators ( Princeton University Press, 1981).Google Scholar
  27. [Tay2]
    M. E. Taylor, Pdo and Non Linear PDE (Birkhauser, Boston, 1991).Google Scholar
  28. [Tay3]
    M. E. Taylor, Partial Differential Equations, Vols. I, II, and III. Appl. Math. Sc., Vols. 115, 116, and 117 (Springer Verlag, 1996).Google Scholar
  29. [Tri1]
    H. Triebel, Theory of Functions Spaces, Vol. I. Monog. in Maths., Vol. 78 (Birkhauser, 1983).Google Scholar
  30. [Tri2]
    H. Triebel, Theory of Functions Spaces, Vol. II. Monog. in Maths., Vol. 84 (Birkhauser, 1992).Google Scholar
  31. [Tri3]
    H. Triebel, Interpolation Theory, Functions Spaces, Differential Operators (North-Holland, Amsterdam, 1978).Google Scholar
  32. [Vil1]
    C. Villani, Thèse Université Paris-Dauphine, Paris (1998).Google Scholar
  33. [Vil2]
    C. Villani, Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off, Rev. Mat. Iberoam, to appear.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Radjesvarane Alexandre
    • 1
  1. 1.MAPMO UMR 6628, Département de MathématiquesUniversité d'OrléansOrleans Cedex 2France

Personalised recommendations