Selected topics in transport phenomena: deterministic and probabilistic aspects

Landau damping and nonlinear echoes

Jacob Bedrossian

University of Maryland

In this talk we will discuss some of the intricacies of Landau damping in the collisionless Vlasov equations and in the collisionless limits of Vlasov-Fokker-Planck equations. We will discuss the construction of solutions to the Vlasov-Poisson equations on S x R which are arbitrarily close to homogeneous equilibrium in Sobolev regularity but which display arbitrarily long sequences of nonlinear oscillations known as plasma echoes. In particular, these oscillations show that Landau's collisionless linearization is not accurate for long times in Sobolev regularity. Further, we show that the inclusion of weak collisional effects can suppress these plasma echoes and make it possible to obtain Sobolev regularity results. Combined with the existing infinite regularity results of Mouhot and Villani, these results together confirm and refute a variety of conjectures made by both mathematicians and physicists over the years regarding Landau damping near homogeneous equilibrium.