Enlarging the functional space of decay estimates on semigroups

TL;DR

This paper introduces a generalized method to expand the functional space for decay estimates of semigroups, demonstrated through applications to Boltzmann, inelastic collisions, and Fokker-Planck equations.

Contribution

It presents a new abstract framework for enlarging the functional space of spectral-gap-like decay estimates, extending previous specific methods.

Findings

Method successfully applied to Fokker-Planck equation

Enables broader spectral gap estimates in kinetic equations

Provides a unified abstract approach for various kinetic models

Abstract

This note briefly presents a new method for enlarging the functional space of a "spectral-gap-like" estimate of exponential decay on a semigroup. A particular case of the method was first devised in hal-00076709 for the spatially homogeneous Boltzmann equation, and a variant was used in Ref. hal-00124876 in the same context for inelastic collisions. We present a generalized abstract version of it, a short proof of the algebraic core of the method, and a new application to the Fokker-Planck equation. More details and other applications shall be found in the work in preparation Ref. [3] (another application to quantum kinetic theory can be found in the work in preparation Ref. [4]).