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 \({r\in (-1,2)}\) 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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The second author of this work was supported by a grant of the Agence Nationale de la Recherche numbered ANR-08-BLAN-0220-01.
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Bally, V., Fournier, N. Regularization properties of the 2D homogeneous Boltzmann equation without cutoff. Probab. Theory Relat. Fields 151, 659–704 (2011). https://doi.org/10.1007/s00440-010-0311-x
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DOI: https://doi.org/10.1007/s00440-010-0311-x