Brief description:

In this course, we review known and discuss open results on some classical kinetic models such as the Boltzmann or the Vlasov equations. We begin with Boltzmann's equation and present the known existence results and the various open issues.

In order to present proofs, we shall review various tools of independent interest such as velocity averaging, the regularity of some part of the Boltzmann collision operator and its connections with Radon transforms, the propagation of SL¹ compactness forward and backward in time and regularization effects due to grazing collisions. We then discuss Vlasov equations such as Vlasov-Poisson and Vlasov-Maxwell systems, and explain how global weak solutions can be built using velocity averaging and (or) the theory of generalized a.e. flows. We conclude by discussing coupled Vlasov-Boltzmann systems.

Prerequisite:  No permission is required from the instructor.