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.Comment: Expository note for the Proceedings of the Journees EDP 2017, based on a talk given at Journees EDP 2017 in Roscoff, France. Aimed at mathematicians who study the analysis of PDEs, but not necessarily those who work specifically on kinetic theory. 16 page

Stable mixing estimates in the infinite Péclet number limit

We consider a passive scalar $f$ advected by a strictly monotone shear flow and with a diffusivity parameter $\nu\ll 1$. We prove an estimate on the homogeneous $\dot{H}^{-1}$ norm of $f$ that combines both the $L^2$ enhanced diffusion effect at a sharp rate proportional to $\nu^{1/3}$, and the sharp mixing decay proportional to $t^{-1}$ of the $\dot{H}^{-1}$ norm of $f$ when $\nu=0$. In particular, the estimate is stable in the infinite P\'eclet number limit, as $\nu\to 0$. To the best of our knowledge, this is the first result of this kind since the work of Kelvin in 1887 on the Couette flow. The two key ingredients in the proof are an adaptation of the hypocoercivity method and the use of a vector field $J$ that commutes with the transport part of the equation. The $L^2$ norm of $Jf$ together with the $L^2$ norm of $f$ produces a suitable upper bound for the $\dot{H}^{-1}$ norm of the solution that gives the extra decay factor of $t^{-1}$

Metastability for the dissipative quasi-geostrophic equation and the non-local enhancement

In this paper, we study the metastability for the 2-D linearized dissipative quasi-geostrophic equation with small viscosity $\nu$ around the quasi steady state $\theta_{sin}=e^{-\nu t}\sin y$. We proved the linear enhanced dissipation and obtained the dissipation rate. Moreover, the new non-local enhancement phenomenon was discovered and discussed. Precisely we showed that the non-local term re-enhances the enhanced diffusion effect by the shear-diffusion mechanism