An introduction to kinetic equations: The Vlasov-Poisson system and the Boltzmann equation

The purpose of kinetic equations is the description of dilute particle gases at an intermediate scale between the microscopic scale and the hydrodynamical scale. By dilute gases, one has to understand a system with a large number of particles, for which a description of the position and of the velocity of each particle is irrelevant, but for which the decription cannot be reduced to the computation of an average velocity at any time t ∈ IR and any position x ∈ IRd: one wants to take into account more than one possible velocity at each point, and the description has therefore to be done at the level of the phase space – at a statistical level – by a distribution function f(t,x,v). This course is intended to make an introductory review of the literature on kinetic equations. Only the most important ideas of the proofs will be given. The two main examples we shall use are the Vlasov-Poisson system and the Boltzmann equation in the whole space