A new regularization possibility for the Boltzmann equation with soft potentials

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September 2008, 1(3): 405-414. doi: 10.3934/krm.2008.1.405

A new regularization possibility for the Boltzmann equation with soft potentials

1.	LAMA UMR 8050, Faculté de Sciences et Technologies, Université Paris Est, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

Received June 2008 Revised June 2008 Published August 2008

Abstract
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We consider a simplified Boltzmann equation: spatially homogeneous, two-dimensional, radially symmetric, with Grad's angular cutoff, and linearized around its initial condition. We prove that for a sufficiently singular velocity cross section, the solution may become instantaneously a function, even if the initial condition is a singular measure. To our knowledge, this is the first regularization result in the case with cutoff: all the previous results were relying on the non-integrability of the angular cross section. Furthermore, our result is quite surprising: the regularization occurs for initial conditions that are not too singular, but also not too regular. The objective of the present work is to explain that the singularity of the velocity cross section, which is often considered as a (technical) obstacle to regularization, seems on the contrary to help the regularization.

Keywords: Boltzmann equation, Grad's cutoff., soft potentials, regularization.

Mathematics Subject Classification: Primary: 76P05, 82C4.

Citation: Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic & Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405

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