On the Boltzmann Equation for Diffusively Excited Granular Media |
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Authors: | IM Gamba V Panferov C Villani |
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Institution: | (1) Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712-1082, USA;(2) UMPA, ENS Lyon, 46 allée d Italie, 69364 Lyon Cedex 07, France;(3) Department of Mathematics and Statistics, University of Victoria, Victoria B.C. V8W 3P4, Canada |
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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 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. |
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