A brief summary of nonlinear echoes and Landau damping

In this expository note we review some recent results on Landau damping in the nonlinear Vlasov equations, focusing specifically on the recent construction of nonlinear echo solutions by the author [arXiv:1605.06841] and the associated background. These solutions show that a straightforward extension of Mouhot and Villani's theorem on Landau damping to Sobolev spaces on $\mathbb T^n_x \times \mathbb R^n_v$ is impossible and hence emphasize the subtle dependence on regularity of phase mixing problems. This expository note is specifically aimed at mathematicians who study the analysis of PDEs, but not necessarily those who work specifically on kinetic theory. However, for the sake of brevity, this review is certainly not comprehensive.