Quantitative Geometric Control in Linear Kinetic Theory

TL;DR

This paper establishes quantitative exponential stabilization estimates for linear kinetic equations with various transport and collision operators, under geometric control conditions, using novel trajectory-based and weighted functional inequality techniques.

Contribution

It introduces a new approach relying on trajectories and weighted inequalities to prove spectral gaps for kinetic equations with diverse boundary and collision conditions.

Findings

Spectral gap estimates are obtained under geometric control conditions.

We develop new weighted functional inequalities for divergence operators.

We show weaker control conditions suffice in hypoelliptic cases.

Abstract

We consider general linear kinetic equations combining transport and a linear collision on the kinetic variable with a spatial weight that can vanish on part of the domain. The considered transport operators include external potential forces and boundary conditions, e.g. specular, diffusive and Maxwell conditions. The considered collision operators include the linear relaxation (scattering) and the Fokker-Planck operators and the boundary conditions include specular, diffusive and Maxwell conditions. We prove quantitative estimates of exponential stabilisation (spectral gap) under a geometric control condition. The argument is new and relies entirely on trajectories and weighted functional inequalities on the divergence operators. The latter functional inequalities are of independent interest and imply quantitatively weighted Stokes and Korn inequalities. We finally show that uniform control conditions are not always necessary for the existence of a spectral gap when the equation is hypoelliptic, and prove weaker control conditions in this case.