Factorization for non-symmetric operators and exponential H-theorem

By Maria Pia Gualdani, St'ephane Mischler, Clément Mouhot. 2010

209 citations recorded by the source corpus.

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 semi- group to study perturbative solutions for the nonlinear Boltzmann equation. We introduce a non-symmetric energy method to prove nonlinear stability in this context in $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, 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.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e61be737ae1a99d5d9759220ab84da89f8cfeff6

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Citations in the corpus

Spectral analysis of semigroups and growth-fragmentation equations (2013, methodology): “Λ = Xn ℓ=0 (−1)ℓ SB ∗(ASB)(∗ℓ) +(−1)n+1 S Λ ∗(ASB)(∗n). It is an usual trick in order to establish eventual norm continuity (see [15] and the references therein) and it has been also recently used in [48, 81] in order to enlarge or to shrink the functional space where the semigroup SΛ satisﬁes some ”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, methodology): “the present paper is twofold: (1) We carry on with developing an abstract method for deriving growth estimates on the semigroup associated to non-symmetric operators in Banach spaces as introduced in [9]. We extend the method so as to consider the shrinkageof the functional space. Roughly speaking, ”