Communications in Mathematical Physics

, Volume 246, Issue 3, pp 503–541 | Cite as

On the Boltzmann Equation for Diffusively Excited Granular Media

  • I.M. Gamba
  • V. Panferov
  • C. Villani


We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative L 2 ( N ) function, with bounded mass and kinetic energy (second moment), we prove the existence of a solution to this model, which instantaneously becomes smooth and rapidly decaying. Under a weak additional assumption of bounded third moment, the solution is shown to be unique. We also establish the existence (but not uniqueness) of a stationary solution. In addition we show that the high-velocity tails of both the stationary and time-dependent particle distribution functions are overpopulated with respect to the Maxwellian distribution, as conjectured by previous authors, and we prove pointwise lower estimates for the solutions.


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  1. 1.
    Arkeryd, L.: On the Boltzmann equation. II. The full initial value problem. Arch. Rational Mech. Anal. 45, 17–34 (1972)MATHGoogle Scholar
  2. 2.
    Benedetto, D., Caglioti, E., Carrillo, J.A., Pulvirenti, M.: A non-Maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91, 979–990 (1998)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Bizon, C., Shattuck, M.D., Swift, J.B., Swinney, H.L.: Transport coefficients for granular media from molecular dynamics simulations. Phys. Rev. E 60, 4340–4351 (1999), ArXiv:cond-mat/9904132CrossRefGoogle Scholar
  4. 4.
    Bobylev, A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Statist. Phys. 88, 1183–1214 (1997)MathSciNetMATHGoogle Scholar
  5. 5.
    Bobylev, A.V., Carrillo, J.A., Gamba, I.M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Statist. Phys. 98, 743–773 (2000)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Bobylev, A.V., Cercignani, C.: Moment equations for a granular material in a thermal bath. J. Statist. Phys. 106, 547–567 (2002)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Bobylev, A.V., Gamba, I.M., Panferov, V.: Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions. In preparationGoogle Scholar
  8. 8.
    Brilliantov, N.V., Poeschel, T.: Granular gases with impact-velocity dependent restitution coefficient. In: Granular Gases, T. Poeschel, S. Luding (eds.), Lecture Notes in Physics, Vol. 564, Berlin: Springer, 2000, pp. 100–124, ArXiv:cond-mat/0204105Google Scholar
  9. 9.
    Carrillo, J.A., Cercignani, C., Gamba, I.M.: Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E (3), 62, 7700–7707 (2000)Google Scholar
  10. 10.
    Cercignani, C.: Recent developments in the mechanics of granular materials. In: Fisica matematica e ingegneria delle strutture, Bologna: Pitagora Editrice, 1995, pp. 119–132Google Scholar
  11. 11.
    Cercignani, C., Illner, R., Stoica, C.: On diffusive equilibria in generalized kinetic theory. J. Statist. Phys. 105, 337–352 (2001)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Desvillettes, L.: Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Rational Mech. Anal. 123, 387–404 (1993)MathSciNetMATHGoogle Scholar
  13. 13.
    Desvillettes, L., Villani, C.: On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Comm. Partial Diff. Eqs. 25, 179–259 (2000)Google Scholar
  14. 14.
    Di Blasio, G.: Differentiability of spatially homogeneous solutions of the Boltzmann equation in the non Maxwellian case. Commun. Math. Phys. 38, 331–340 (1974)MATHGoogle Scholar
  15. 15.
    Elmroth, T.: Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Ra. Mech. Anal. 82, 1–12 (1983)MathSciNetMATHGoogle Scholar
  16. 16.
    Ernst, M.H., Brito, R.: Velocity tails for inelastic Maxwell models (2001), ArXiv:cond-mat/0111093Google Scholar
  17. 17.
    Ernst, M.H., Brito, R.: Scaling solutions of inelastic Boltzmann equations with over-populated high energy tails. J. Stat. Phys., to appear (2002), ArXiv:cond-mat/0112417Google Scholar
  18. 18.
    Gamba, I.M., Panferov, V., Villani, C.: Work in progress Google Scholar
  19. 19.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Vol. 224 of Grundlehren der Mathematischen Wissenschaften, Berlin: Springer-Verlag, second~ed., 1983Google Scholar
  20. 20.
    Goldhirsch, I.: Rapid granular flows: kinetics and hydrodynamics. In: Modeling in applied sciences, Model. Simul. Sci. Eng. Technol., Boston, MA: Birkhäuser Boston, 2000, pp. 21–79Google Scholar
  21. 21.
    Goldshtein, A., Shapiro, M.: Mechanics of collisional motion of granular materials. I. General hydrodynamic equations. J. Fluid Mech. 282, 75–114 (1995)MATHGoogle Scholar
  22. 22.
    Gustafsson, T.: Global L p-properties for the spatially homogeneous Boltzmann equation. Arch. Rational Mech. Anal. 103, 1–38 (1988)MATHGoogle Scholar
  23. 23.
    Jenkins, J.T.: Kinetic theory for nearly elastic spheres. In: Physics of dry granular media (Cargèse, 1997), NATO Adv. Sci. Inst. Ser. E Appl. Sci., 350, Dordrecht: Kluwer Acad. Publ., 1998, pp. 353–369Google Scholar
  24. 24.
    Jenkins, J.T., Richman, M.W.: Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Rational Mech. Anal. 87, 355–377 (1985)MathSciNetMATHGoogle Scholar
  25. 25.
    Jenkins, J.T., Savage, S.B.: A theory for rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202 (1983)MATHGoogle Scholar
  26. 26.
    Krapivsky, P.L., Ben-Naim, E.: Multiscaling in infinite dimensional collision processes. Phys. Rev. E 61, R5 (2000), ArXiv:cond-mat/9909176Google Scholar
  27. 27.
    Krapivsky, P.L., Ben-Naim, E.: Nontrivial velocity distributions in inelastic gases. J. Phys. A 35, L147 (2002), ArXiv:cond-mat/0111044Google Scholar
  28. 28.
    Kudrolli, A., Henry, J.: Non-Gaussian velocity distributions in excited granular matter in the absence of clustering. Phys. Rev. E 62, R1489 (2000), ArXiv:cond-mat/0001233Google Scholar
  29. 29.
    Ladyzhenskaya, O., Uraltseva, N., Solonnikov, V.: Linear and quasi-linear equations of parabolic type. Vol. 23 of AMS Translations of Mathematical Monographs, Providence, RI: Am. Math. Soc., 1988Google Scholar
  30. 30.
    Losert, W., Cooper, D., Delour, J., Kudrolli, A., Gollub, J.: Velocity statistics in excited granular media. Chaos 9, 682–690 (1999), ArXiv:cond-mat/9901203CrossRefMATHGoogle Scholar
  31. 31.
    Lu, X.: Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation. J. Stat. Phys. 96, 765–796 (1999)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non-Linéaire 16, 467–501 (1999)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Moon, S.J., Shattuck, M.D., Swift, J.B.: Velocity distributions and correlations in homogeneously heated granular media. Phys. Rev. E 64, 031303–1–031303–10 (2001) ArXiv:cond-mat/0105322CrossRefGoogle Scholar
  34. 34.
    Mouhot, C., Villani, C.: Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. To appear in Archive for Rational Mechanics and AnalysisGoogle Scholar
  35. 35.
    Povzner, A.J.: On the Boltzmann equation in the kinetic theory of gases. Mat. Sb. (N.S.) 58(100), 65–86 (1962)MATHGoogle Scholar
  36. 36.
    Rouyer, F., Menon, N.: Velocity fluctuations in a homogeneous 2d granular gas in steady state. Phys. Rev. Lett. 85, 3676 (2000)CrossRefGoogle Scholar
  37. 37.
    Umbanhowar, P.B., Melo, F., Swinney, H.L.: Localized excitations in a vertically vibrated granular layer. Nature 382, 793–796 (1996)CrossRefGoogle Scholar
  38. 38.
    van Noije, T., Ernst, M.: Velocity distributions in homogeneously cooling and heated granular fluids. Gran. Matt. 1, 57–8 (1998), ArXiv:cond-mat/9803042CrossRefGoogle Scholar
  39. 39.
    Villani, C.: A survey of mathematical topics in collisional kinetic theory. In: Handbook of Mathematical Fluid Mechanics, Friedlander, S., Serre, D. (eds.), Vol. 1, Amsterdam: Elsevier, 2002, ch. 2Google Scholar
  40. 40.
    Wennberg, B.: Entropy dissipation and moment production for the Boltzmann equation. J. Statist. Phys. 86, 1053–1066 (1997)MathSciNetMATHGoogle Scholar
  41. 41.
    Williams, D.R.M., MacKintosh, F.C.: Driven granular media in one dimension: Correlations and equation of state. Phys. Rev. E 54, 9–12 (1996)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • I.M. Gamba
    • 1
  • V. Panferov
    • 1
    • 3
  • C. Villani
    • 2
  1. 1.Department of MathematicsThe University of Texas at AustinUSA
  2. 2.UMPAENS LyonLyon Cedex 07France
  3. 3.Department of Mathematics and StatisticsUniversity of VictoriaCanada

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