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1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision 2 HARVARD - Department of Mathematics Cambridge

Abstract : 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 $\phir = r^{-s−1}$, $s \ge 5$ or the so-called moderately soft potentials $\phir = r^{−s−1}$, $3 < s < 5$, 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.

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





Autor: Clément Mouhot - Robert Strain -

Fuente: https://hal.archives-ouvertes.fr/



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