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Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

Abstract : We consider the 2-dimensional spatially homogeneous Boltzmann equation for hard potentials. We assume that the initial condition is a probability measure that has some exponential moments and is not a Dirac mass. We prove some regularization properties: for a class of very hard potentials, the solution instantaneously belongs to H(r) , for some r is an element of (-1, 2) depending on the parameters of the equation. Our proof relies on the use of a well-suited Malliavin calculus for jump processes.
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https://hal-upec-upem.archives-ouvertes.fr/hal-00692754
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Submitted on : Tuesday, May 1, 2012 - 12:50:46 PM
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Vlad Bally, Nicolas Fournier. Regularization properties of the 2D homogeneous Boltzmann equation without cutoff. Probability Theory and Related Fields, Springer Verlag, 2011, 151 (3-4), pp.659--704. ⟨10.1007/s00440-010-0311-x⟩. ⟨hal-00692754⟩

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