Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

Details Regularization properties of the 2D homogeneous Boltzmann equation without cutoff Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

2009-11-13

arxiv.org/abs/0911.2614v2

Mathematics

Probability

Scientific paper

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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