Factorization for non-symmetric operators and exponential H-theorem

TL;DR

This paper develops a novel abstract factorization method for non-symmetric operators to derive decay estimates, and applies it to kinetic equations, culminating in a constructive proof of exponential decay to equilibrium for the nonlinear Boltzmann equation.

Contribution

Introduces a high-order factorization approach for non-symmetric operators and applies it to kinetic equations, proving exponential decay to equilibrium in the nonlinear Boltzmann context.

Findings

Established decay estimates for non-symmetric operators in Banach spaces.

Proved exponential convergence to equilibrium for the nonlinear Boltzmann equation.

Provided the first constructive proof of sharp exponential decay in this setting.

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 $L^1_v L^\infty _x (1 + |v|^k)$, $k > 2$, 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.