21808 articles – 15604 references  [version française]
 HAL: hal-00349854, version 3
 arXiv: 0909.1229
 Available versions: v1 (2009-01-05) v2 (2009-09-07) v3 (2009-12-10)
 Regularizing effect and local existence for non-cutoff Boltzmann equation
 Radjesvarane Alexandre 1, Y. Morimoto 2
 (2008-12-24)
 The Boltzmann equation without Grad's angular cutoff assumption is believed to have regularizing effect on the solution because of the non-integrable angular singularity of the cross-section. However, even though so far this has been justified satisfactorily for the spatially homogeneous Boltzmann equation, it is still basically unsolved for the spatially inhomogeneous Boltzmann equation. In this paper, by sharpening the coercivity and upper bound estimates for the collision operator, establishing the hypo-ellipticity of the Boltzmann operator based on a generalized version of the uncertainty principle, and analyzing the commutators between the collision operator and some weighted pseudo differential operators, we prove the regularizing effect in all (time, space and velocity) variables on solutions when some mild regularity is imposed on these solutions. For completeness, we also show that when the initial data has this mild regularity and Maxwellian type decay in velocity variable, there exists a unique local solution with the same regularity, so that this solution enjoys the $C^\infty$ regularity for positive time.
 1: Institut de Recherche de l'Ecole Navale (EA 3634) (IRENAV) Ecole Navale – Arts et Métiers ParisTech 2: Graduate School of Human and Environmental Studies Kyoto University 3: Liu Bie Ju Centre for Mathematical Sciences City University Of Hong-Kong 4: Laboratoire de Mathématiques Raphaël Salem (LMRS) CNRS : UMR6085 – Université de Rouen 5: Departement of Mathematics City University of Hong Kong
 Subject : Mathematics/Analysis of PDEs
 Keyword(s): Non-cutoff cross-sections – Boltzmann equation – regularizing effect – local existence – uncertainty principle – pseudo-differential calculus
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 hal-00349854, version 3 http://hal.archives-ouvertes.fr/hal-00349854 oai:hal.archives-ouvertes.fr:hal-00349854 From: Radjesvarane Alexandre <> Submitted on: Thursday, 10 December 2009 14:21:29 Updated on: Thursday, 10 December 2009 16:28:57