Sanchit Chaturvedi (Stanford), Jonathan Luk (Stanford), and I just submitted the paper “The Vlasov–Poisson–Landau system in the weakly collisional regime”, where we prove Landau damping and extra dissipation for plasmas modeled by the physical Vlasov-Poisson-Landau system in the weakly collisional regime 
Vlasov-Poisson
Plasma echoes near stable Penrose data
Echoes in a plasma are the excitement of new waves due to nonlinear interaction. The excitement may happen at an arbitrarily large time, which is the main source of difficulties in understanding Landau damping. For analytic data, the echoes are suppressed as the electric field is exponentially localized in time, and the nonlinear Landau damping holds for such data, as was first obtained by Mouhot and Villani in their celebrated work (Acta Math 2011; see also the extension to include Gevrey data). The nonlinear Landau damping remains largely elusive for less regular data (e.g., data with Sobolev regularity).
Recently, in a collaboration with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), we give an elementary proof of the known Landau damping results, which I also blogged it here, that were seen as a perturbation of the free transport dynamics, whose damping is direct (that is, the phase mixing). In the companion paper with E. Grenier and I. Rodnianski, we construct a class of echo solutions, which are arbitrarily large in any Sobolev spaces (in particular, they do not belong to the analytic or Gevrey classes studied by Mouhot and Villani), but nonetheless, the nonlinear Landau damping holds. In this blog post, I shall briefly discuss the plasma echo mechanism and our new results.
Landau damping for analytic and Gevrey data
Landau damping is a classical subject in Plasma Physics, which studies decay of the electric field in a collisionless plasma in the large time. The damping was discovered and fully understood by Landau in the 40s for the linearized evolution near Maxwellians, and later extended by O. Penrose in the 60s for general spatially homogenous equilibria. The first mathematical proof of the nonlinear Landau damping was given by Mouhot and Villani for analytic data in their celebrated work (Acta Math, 2011). Their proof was then simplified, and the result was extended by Bedrossian, Masmoudi, and Mouhot to include data in certain Gevrey classes (Annals of PDEs, 2016).
Recently, in a collaboration with E. Grenier (ENS Lyon) and I. Rodnianski (Princeton), we give an elementary proof of these same results, which I shall give a sketch of it in this blog post. To avoid some tedious algebra, I mainly focus on the analytic case, which is precisely the case originally studied by Mouhot and Villani, leaving some remarks to the Gevrey cases at the very end of the post, where you’ll also find the slides of my recent lectures over Zoom on this topics.
Landau damping for screened Vlasov-Poisson on the whole space
In a recent joint work with Daniel Han-Kwan (CMLS, Ecole polytechnique) and Frédéric Rousset (Paris-Sud University), we give an alternative proof of the Landau damping for screened Vlasov-Poisson system near stable homogenous equilibria on the whole space, a result that was first established by Bedrossian, Masmoudi and Mouhot, for data with finite Sobolev regularity (they remarked that 36 derivatives were sufficient).
Bardos-Degond’s solutions to Vlasov-Poisson
This is part of my lecture notes on Kinetic Theory of Gases, taught at Penn State last semester, Fall 2017. In this part, I’d like to introduce this nice Bardos-Degond 1985’s global solutions to the Vlasov-Poisson system. Of course, the global smooth solutions are already constructed, without any restriction on size of initial data (e.g., Pfaffelmoser, Schaeffer ’91; see also the previous lecture), however they give no information on their asymptotic behavior at large time. Now, for initial data that are sufficiently small near zero, Bardos and Degond were able to construct global smooth solutions that decay in large time. To my knowledge, this was the first result where dispersion is rigorously shown for kinetic equations (they appear to be motivated by similar results for nonlinear wave equations where dispersion was (still is) the key to deduce the global behavior at the large time; e.g., Klainerman, Ponce, Shatah, among others, in the early 80s).
On the non-relativistic limit of Vlasov-Maxwell
In this note, I briefly explain my recent joint work with D. Han-Kwan (CNRS, Ecole polytechnique) and F. Rousset (Paris-Sud) on the non-relativistic limit of Vlassov-Maxwell. Precisely, we consider the relativistic Vlasov-Maxwell system, modeling the dynamics of electrons with electron density distribution 
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Kinetic theory: global solution to 3D Vlasov-Poisson
One of the beautiful results in kinetic theory is to construct the global classical solution to the 3D Vlasov-Poisson system. The result is now classical; see, for instance, chapter 4 of Glassey‘s book. However, I feel the result is a bit non-trivial to convey to students and beginners. Would you agree? Anyway, this post is to try to present this classical result, aiming to be as pedagogical as possible, with the original the good, the bad, and the ugly proof of J. Schaeffer ’91.