Landau damping: Gevrey regularity and paraproducts

We present the key ideas of a new proof of Landau damping for the Vlasov-Poisson equation obtained in a joint work with Bedrossian and Masmoudi. This nonlinear transport equation is a fundamental model for describing self-interacting plasmas or galaxies, and Landau damping is a nonlinear stability mechanism based on phase mixing. The new method does not use a Newton scheme and seems to capture a critical Gevrey regularity for the damping. Moreover its greater flexibility should open the way to further work, and we will finish by sketching some open questions.