Communications in Mathematical Physics

, Volume 246, Issue 3, pp 503-541

First online:

On the Boltzmann Equation for Diffusively Excited Granular Media

  • I.M. GambaAffiliated withDepartment of Mathematics, The University of Texas at Austin
  • , V. PanferovAffiliated withDepartment of Mathematics, The University of Texas at AustinDepartment of Mathematics and Statistics, University of Victoria
  • , C. VillaniAffiliated withUMPA, ENS Lyon

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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.