Factorization of Non-Symmetric Operators and Exponential H-Theorem

Abstract

We present an abstract method for deriving decay estimates on the resolvents and semigroups of non-symmetric operators in Banach spaces in terms of estimates in another smaller reference Banach space. This applies to a class of operators writing as a regularizing part, plus a dissipative part. The core of the method is a high-order quantitative factorization argument on the resolvents and semigroups. We then apply this approach to the Fokker-Planck equation, to the kinetic Fokker-Planck equation in the torus, and to the linearized Boltzmann equation in the torus.We finally use this information on the linearized Boltzmann semigroup to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in L 1 v L ∞ x (1 + |v| k ), k > 2, with sharp rate of decay in time.As a consequence of these results we obtain the first constructive proof of exponential decay, with sharp rate, towards global equilibrium for the full nonlinear Boltzmann equation for hard spheres, conditionally to some smoothness and (polynomial) moment estimates. This improves the result in [32] where polynomial rates at any order were obtained, and solves the conjecture raised in [91,29,86] about the optimal decay rate of the relative entropy in the H-theorem. (2000): 47D06 One-parameter semigroups and linear evolution equations [See also 34G10, 34K30], 35P15 Estimation of eigenvalues, upper and lower bounds, 47H20 Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07], 35Q84 Fokker-Planck equations, 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05]. Mathematics Subject Classification

Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting

“…The proof relies on the combination of Proposition 2.14 and the Theorem 2.13 of enlargement on the functional space of semigroup decay from [14]. Our small space is E and our large space is E. The assumptions of the theorem are clearly fulfilled thanks to Lemmas 2.8 and 2.9 and thus yield the conclusion.…”

“…It is thus not possible to develop a perturbative theory around the elastic case in this kind of spaces. We thus have to use the recent work [14] in which the study of the linearized elastic Boltzmann operator has been developed in larger Banach spaces, it gives explicit spectral gap estimates on the semigroup associated to the linearized non homogeneous operator L 0 in various Sobolev spaces W s,p x W σ,q v (m) with polynomial or stretched exponential weights m. In this kind of spaces which can contain the inelastic equilibriums G λ , one could consider to develop a perturbative argument around the elastic case. Here is a rough version of the main result that we obtain on the linearized operator (see Theorem 2.21 for a complete version):…”

“…The first one comes from [14]: the spectrum of the linearized elastic equation is well localized, meaning that it admits a spectral gap in a large class of Sobolev spaces. The second one is that for λ small enough, L λ − L 0 = O(λ) for a suitable operator norm, this kind of estimates relies on accurate estimates on the difference Q e λ − Q 1 that we establish.…”

“…We obtain a new result on it, we prove existence of a spectral gap in weighted (stretched exponential and polynomial) Sobolev spaces and a result of exponential stability for the semigroup generated by the linearized operator. To do that, we develop a perturbative argument around the spatially inhomogeneous equation for elastic hard spheres and we take advantage of the recent paper [14] where this equation has been considered. We then link the linearized theory with the nonlinear one in order to handle the full non-linear problem thanks to new bilinear estimates on the collision operator that we establish.…”

Asymptotic behaviour of the Landau equation with Coulomb potential

“…This enlargement trick we develop in [7] is inspired on the extension theory of [18] (introduced in [22]) and it generalises the theory of [18] to the case in which the operator does not possesses a spectral gap (hence its associated semigroup is not exponentially stable).…”