HAL : hal-00086958, version 2

arXiv : math.AP/0607495

DOI : 10.1016/j.matpur.2007.03.003

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Journal de Mathématiques Pures et Appliqués 87, 5 (2007) 515-535

Versions disponibles :

v1 (20-07-2006)

v2 (21-07-2006)

Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff

Clément Mouhot 1, Robert Strain 2

(05/2007)

In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials $\phi(r) = r^{-(s−1)}$, $s \ge 5$ or the so-called moderately soft potentials $\phi(r) = r^{−(s−1)}$, $3 < s < 5$, (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao [46]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [34] and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators.

1 :	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine
2 :	Department of Mathematics (HARVARD)
	University of Harvard

Domaine

:

Mathématiques/Equations aux dérivées partielles Mathématiques/Physique mathématique Physique/Physique mathématique

Mots Clés : coercivity estimates – linearized Boltzmann operator – linearized Landau operator – quantitative – long-range interaction – non-cutoff – soft potentials – spectral gap

hal-00086958, version 2

http://hal.archives-ouvertes.fr/hal-00086958

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Contributeur : Clément Mouhot <>

Soumis le : Vendredi 21 Juillet 2006, 13:49:59

Dernière modification le : Mardi 18 Décembre 2007, 15:52:59