Communications in Mathematical Physics

, Volume 246, Issue 3, pp 503–541

On the Boltzmann Equation for Diffusively Excited Granular Media

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

DOI: 10.1007/s00220-004-1051-5

Cite this article as:
Gamba, I., Panferov, V. & Villani, C. Commun. Math. Phys. (2004) 246: 503. doi:10.1007/s00220-004-1051-5

Abstract

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 L2(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.

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