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

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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 ϕ ( r ) = r − ( s − 1 ) , s ⩾ 5 , or the so-called moderately soft potentials ϕ ( 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 [Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. I, Comm. Pure Appl. Math. 27 (1974) 407–428; Y.P. Pao, Boltzmann collision operator with inverse-power intermolecular potentials. II, Comm. Pure Appl. Math. 27 (1974) 559–581]. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in [Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002) 391–434] 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.