Abstract:

The Landau equation is an approximation of Boltzmann's kinetic equation for dilute gas in the case of Coulomb interactions between the particles (a setting where the latter equation fails to make sense), mainly used as a model for collisional plasmas. The formal derivation of both the Boltzmann and Landau equations relies crucially on the hypothesis of so-called molecular chaos, roughly stating that particles become independent of one another as their number becomes infinite. It asks for a rigorous justification starting from microscopic N-particle systems, a process now known as propagation of chaos.

In this talk, I will discuss tightness/uniqueness methods for propagation of chaos, which consist in showing tightness (compactness) of the N-particle system, and then that any limit point of the system as N goes to infinity is the unique solution of the Landau equation. Although tightness is relatively straightforward thanks to the recent progress on the links between the Landau equation and Fisher information, uniqueness is a more delicate task. It will lead us to discuss the different notions of solution to the Landau equation, as well as new properties in infinite dimension of second-order Fisher information.