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

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.