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On the Boltzmann Equation for Diffusively Excited Granular Media

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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 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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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712-1082, USA

    I.M. Gamba & V. Panferov

  2. UMPA, ENS Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    C. Villani

  3. Department of Mathematics and Statistics, University of Victoria, Victoria B.C. V8W 3P4, Canada

    V. Panferov

Authors
  1. I.M. Gamba
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  2. V. Panferov
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  3. C. Villani
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Communicated by H.-T. Yau

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Gamba, I., Panferov, V. & Villani, C. On the Boltzmann Equation for Diffusively Excited Granular Media. Commun. Math. Phys. 246, 503–541 (2004). https://doi.org/10.1007/s00220-004-1051-5

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  • DOI: https://doi.org/10.1007/s00220-004-1051-5

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