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

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 suucient to prove the convergence of the solution to a unique stationary solution, as time goes to innnity. The method is also used to analyze other types of boundary conditions such as mass and energy preserving diiuse reeection boundary conditions, and to prove the uniqueness of stationary solutions for some special collision terms. irreversibility { H-Theorem { injection boundary conditions { diiusive boundary conditions { relative entropy { large time asymptotics { uniqueness { stationary solutions { minimization under constraints { nonlinear stability { Casimir energy { Bolza problem { plasmas { semiconductors. 1 Preliminaries and examples In this paper, we study the large time behaviour of the solutions of the initial-boundary value problem for the Vlasov-Poisson and the Vlasov-Poisson-Boltzmann systems. We consider charged particles (say electrons) described by their distribution function f(x; v; t) where x is the position of the particle, v its velocity and t is the time variable. The position x is assumed to lie in a bounded domain ! of IR d (d = 1; 2; 3). The velocity variable v is in IR d whereas the time variable t is in IR +. We shall assume that the particles are submitted to an electric eld E = E(x; t) deriving from the sum of an external potential 0 (x) and a self-consistent potential (x; t) created by the electrons themselves through the Coulomb interaction. The potential 0 (x) is a given stationary function which can be seen for instance as the result of an applied voltage and a background ion density, whereas is a solution of the Poisson equation with homogeneous Dirichlet boundary conditions on @!. We shall also assume that electrons may suuer collisions modelled by a collision operator Q(f). We consider the Cauchy problem corresponding to an initial distribution function f 0 at time t = 0. Various boundary 1