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Probability Theory and Related Fields

, Volume 151, Issue 3–4, pp 659–704 | Cite as

Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

  • Vlad BallyEmail author
  • Nicolas Fournier
Article

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.

Keywords

Kinetic equations Hard potentials without cutoff Malliavin calculus Jump processes 

Mathematics Subject Classification (2000)

60H07 82C40 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.LAMA UMR 8050Université Paris Est, Cité DescartesMarne la Vallée CedexFrance
  2. 2.LAMA UMR 8050, Faculté de Sciences et TechnologiesUniversité Paris EstCréteil CedexFrance

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