Convergence Rate To Equilibrium For Collisionless Transport Equations With Diffuse Boundary Operators: A New Tauberian Approach

TL;DR

This paper introduces a novel tauberian method to analyze the convergence rate to equilibrium for collisionless transport equations with diffuse boundary operators, achieving near-optimal algebraic decay rates under broad conditions.

Contribution

It develops a new tauberian approach and detailed spectral analysis to determine convergence rates, extending previous methods to more general boundary conditions.

Findings

Achieves near-optimal algebraic convergence rates

Provides a systematic framework for spectral analysis of transport semigroups

Establishes the influence of boundary operator integrability on convergence

Abstract

This paper provides a new tauberian approach to the study of quantitative time asymptotics of collisionless transport semigroups with general diffuse boundary operators. We obtain an (almost) optimal algebraic rate of convergence to equilibrium under very general assumptions on the initial datum and the boundary operator. The rate is prescribed by the maximal gain of integrability that the boundary operator is able to induce. The proof relies on a representation of the collisionless transport semigroups by a (kind of) Dyson-Phillips series and on a fine analysis of the trace on the imaginary axis of Laplace transform of remainders (of large order) of this series. Our construction is systematic and is based on various preliminary results of independent interest.