Factorization for 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 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 $L_v^1{L}_x^{\infty }(1+|v{|}^k)$, $k>2$, 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.

118 pages, v2 includes new results and methods, v3 corrects a typo in the definition of some high-order norms in sections 4 and 5

References in corpus (12)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
• On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates
• Cauchy problem and exponential stability for the inhomogeneous Landau equation
• Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
• From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight
• From resolvent bounds to semigroup bounds
• On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials
• Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations
• Rate of convergence to self-similarity for the fragmentation equation in L^1 spaces
• Exponential convergence to equilibrium for the homogeneous Boltzmann equation for hard potentials without cut-off
• Exponential Time Decay Estimates for the Landau Equation on Torus

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• Global Existence of Non-cutoff Boltzmann Equation in Weighted Sobolev Space
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