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arXiv : 1007.0304

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Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential

Radjesvarane Alexandre 1, Yoshinori Morimoto 2, Seiji Ukai 3, Chao-Jiang Xu 4, Tong Yang 5

(27/10/2010)

It is known that the singularity in the non-cutoff cross-section of the Boltzmann equation leads to the gain of regularity and gain of weight in the velocity variable. By defining and analyzing a non-isotropy norm which precisely captures the dissipation in the linearized collision operator, we first give a new and precise coercivity estimate for the non-cutoff Boltzmann equation for general physical cross sections. Then the Cauchy problem for the Boltzmann equation is considered in the framework of small perturbation of an equilibrium state. In this part, for the soft potential case in the sense that there is no positive power gain of weight in the coercivity estimate on the linearized operator, we derive some new functional estimates on the nonlinear collision operator. Together with the coercivity estimates, we prove the global existence of classical solutions for the Boltzmann equation in weighted Sobolev spaces.

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 :	retaite (Mr.)
	retraité
4 :	Laboratoire de Mathématiques Raphaël Salem (LMRS)
	CNRS : UMR6085 – Université de Rouen
5 :	department of mathematics
	City University Of Hong-Kong

Domaine

:

Mathématiques/Equations aux dérivées partielles

Mots Clés : Boltzmann equation – coercivity estimate – non-cutoff cross sections – global existence – non-isotropic norm – soft potential

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Contributeur : Radjesvarane Alexandre <>

Soumis le : Mercredi 27 Octobre 2010, 08:17:37

Dernière modification le : Mercredi 27 Octobre 2010, 08:31:36