Regularization properties of the 2D homogeneous Boltzmann equation without cutoff |
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| Authors: | Vlad Bally Nicolas Fournier |
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| Affiliation: | 1. LAMA UMR 8050, Universit?? Paris Est, Cit?? Descartes, 5 boulevard Descartes, Champs sur Marne, 77454, Marne la Vall??e Cedex, France
2. LAMA UMR 8050, Facult?? de Sciences et Technologies, Universit?? Paris Est, 61, avenue du G??n??ral de Gaulle, 94010, Cr??teil Cedex, France
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| 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 ${rin (-1,2)}$ 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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