Skip to main content
SpringerLink
Log in

Rate of Convergence to Equilibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

For the spatially homogeneous Boltzmann equation with hard potentials and Grad's cutoff (e.g. hard spheres), we give quantitative estimates of exponential convergence to equilibrium, and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, on which we provide a lower bound. Our approach is based on establishing spectral gap-like estimates valid near the equilibrium, and then connecting the latter to the quantitative nonlinear theory. This leads us to an explicit study of the linearized Boltzmann collision operator in functional spaces larger than the usual linearization setting.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arkeryd, L. On the Boltzmann equation. Arch. Rat. Mech. Anal. 45, 1–34 (1972)

    Google Scholar 

  2. Arkeryd, L.: Stability in L 1 for the spatially homogeneous Boltzmann equation. Arch. Rat. Mech. Anal. 103, 151–167 (1988)

    Google Scholar 

  3. Baranger, C., Mouhot, C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Accepted for publication in Rev. Mat. Iberoamericana

  4. Bobylev, A.V.: The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Mathematical physics reviews, Vol. 7, Soviet Sci. Rev. Sect. C Math. Phys. Rev., Chur: Harwood Acad. Publ., 1988, pp. 111–233

  5. Bobylev, A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88, 1183–1214 (1997)

    Google Scholar 

  6. Bobylev, A.V., Cercignani, C.: On the rate of entropy production for the Boltzmann equation. J. Stat. Phys. 94, 603–618 (1999)

    Google Scholar 

  7. Bouchut, F., Desvillettes, L.: A proof of the smoothing property of the positive part of Boltzmann's kernel. Rev. Math. Iberoam. 14, 47–61 (1998)

    Google Scholar 

  8. Bobylev, A.V., Gamba, I.M., Panferov, V.A.: Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions. J. Stat. Phys. 116, 1651–1682 (2004)

    Google Scholar 

  9. Brezis, H.: Analyse fonctionnelle. Théorie et applications. Paris: Masson, 1983

  10. Caflisch, R.E.: The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous. Commum. Math. Phys. 74, 71–95 (1980)

    Google Scholar 

  11. Carleman, T.: Sur le théorie de l'équation intégrodifférentielle de Boltzmann. Acta Math. 60, 369–424 (1932)

    Google Scholar 

  12. Carlen, E.A., Carvalho, M.C.: Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation. J. Stat. Phys. 67, 575–608 (1992)

    Google Scholar 

  13. Carlen, E.A., Carvalho, M.C.: Entropy production estimates for Boltzmann equations with physically realistic collision kernels. J. Stat. Phys. 74, 743–782 (1994)

    Google Scholar 

  14. Carlen, E.A., Lu, X.: Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums. J. Stat. Phys. 112, 59–134 (2003)

    Google Scholar 

  15. Carlen, E.A., Gabetta, E., Toscani, G.: Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys. 199, 521–546 (1999)

    Google Scholar 

  16. Cercignani, C.: The Boltzmann equation and its applications. Applied Mathematical Sciences 67, New York: Springer-Verlag, 1988

  17. Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. New York: Springer-Verlag, 1994

  18. Ellis, R.S., Pinsky, M.A.: The first and second fluid approximations to the linearized Boltzmann equation. J. Math. Pures Appl. 54(9), 125–156 (1975)

    Google Scholar 

  19. Golse, F., Poupaud, F.: Un résultat de compacité pour l'équation de Boltzmann avec potentiel mou. Application au problème de demi-espace. C. R. Acad. Sci. Paris Sér. I Math. 303, 585–586 (1986)

  20. Grad, H.: Principles of the kinetic theory of gases. In: Flügge's Handbuch des Physik, Vol. XII, Berlin-Heidelberg-New York: Springer-Verlag, 1958, pp. 205–294

  21. Grad, H.: Asymptotic theory of the Boltzmann equation. II. In: Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l'UNESCO, Paris, 1962), Vol. I, 1963, pp. 26–59

  22. Henry, D.: Geometric theory of semilinear parabolic equations. Berlin-New York: Springer-Verlag, 1981

  23. Kato, T.: Perturbation theory for linear operators. Berlin: Springer-Verlag, 1995

  24. Kuscer, I., Williams, M.M.R.: Phys. Fluids 10, 1922 (1967)

  25. Lions, P.-L.: Compactness in Boltzmann's equation via Fourier integral operators and applications I,II,III. J. Math. Kyoto Univ. 34, 391–427, 429–461, 539–584 (1994)

    Google Scholar 

  26. Lu, X.: A direct method for the regularity of the gain term in the Boltzmann equation. J. Math. Anal. Appl. 228, 409–435 (1998)

    Google Scholar 

  27. Mischler, S., Mouhot, C., Rodriguez Ricard, M.: Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem. To appear in J. Stat. Phys.

  28. Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 467–501 (1999)

    Google Scholar 

  29. Mouhot, C., Villani, C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Rat. Mech. Anal. 173, 169–212 (2004)

    Google Scholar 

  30. Toscani, G., Villani, C.: Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. Commum. Math. Phys. 203, 667–706 (1999)

    Google Scholar 

  31. Ukai, S.: On the existence of global solutions of a mixed problem for the nonlinear Boltzmann equation. Proc. Japan Acad. 50, 179–184 (1974)

    Google Scholar 

  32. Villani, C.: Cercignani's conjecture is sometimes true and always almost true. Commun. Math. Phys. 243, 455–490 (2003)

    Google Scholar 

  33. Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, Amsterdam: North-Holland, 2002 pp. 71–305

  34. Wang Chang, C.S., Uhlenbeck, G.E., de Boer, J.: Studies in Statistical Mechanics, Vol. V. Amsterdam: North-Holland, 1970

  35. Wennberg, B.: Stability and exponential convergence in L p for the spatially homogeneous Boltzmann equation. Nonlinear Anal. 20(8), 935–964 (1993)

    Google Scholar 

  36. Wennberg, B.: Regularity in the Boltzmann equation and the Radon transform. Commun. Partial Diff. Eqs. 19, 2057–2074 (1994)

    Google Scholar 

  37. Wennberg, B.: Entropy dissipation and moment production for the Boltzmann equation. J. Stat. Phys. 86, 1053–1066 (1997)

    Google Scholar 

Download references

Author information

Author notes

  1. Clément Mouhot

    Present address: CNRS, CEREMADE, Univ. Paris Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

Authors and Affiliations

  1. UMPA, ÉNS Lyon, 46 allée d'Italie, 69364, Lyon Cedex 07, France

    Clément Mouhot

Authors
  1. Clément Mouhot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clément Mouhot.

Additional information

Communicated by J.L. Lebowitz

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mouhot, C. Rate of Convergence to Equilibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials. Commun. Math. Phys. 261, 629–672 (2006). https://doi.org/10.1007/s00220-005-1455-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-005-1455-x

Keywords

Search

Navigation