Relative Entropies for Kinetic Equations in Bounded Domains (Irreversibility, Stationary Solutions, Uniqueness)

Abstract

The relative-entropy method describes the irreversibility of the Vlasov-Poisson and Vlasov-Boltzmann-Poisson systems in bounded domains with incoming boundary conditions. Uniform-in-time estimates are deduced from the entropy. In some cases, these estimates are sufficient to prove the convergence of the solution to a unique stationary solution, as time goes to infinity. The method is also used to analyse other types of boundary conditions such as mass- and energy-preserving diffuse-reflection boundary conditions, and to prove the uniqueness of stationary solutions for some special collision terms.