Abstract

In this talk we will introduce the spatially homogeneous Landau equation for Maxwellian molecules, widely studied by Villani and Desvillettes, among others. It is a non-linear partial differential equation where the unknown function is the density of a gas in the phase space of all positions and velocities of particles. This equation is a common kinetic model in plasma physics and is obtained as a limit of the Boltzmann equation, when all the collisions become grazing. We will first recall some known results. Namely, the existence and uniqueness of the solution to this PDE , as well as its probabilistic interpretation in terms of a non-linear diffusion due to Gurin. We will then show how to obtain Gaussian lower and upper bound for the solution via probabilistic techniques. Joint work with Franois Delarue and Stphane Menozzi.