Dafermos and Rodnianski introduced an 
Year: 2018
Mourre’s theory and local decay estimates, with some applications to linear damping in fluids
In his famous 1981 paper, Mourre gave a sufficient condition for a self-adjoint operator 
![\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+P_I+i%5BH%2CA%5D+P_I+%5Cge+%5Ctheta_I+P_I+%2B+P_I+K+P_I+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0 "\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)")
for some positive constant 
![{[H,A] = HA - AH}](https://s0.wp.com/latex.php?latex=%7B%5BH%2CA%5D+%3D+HA+-+AH%7D&bg=ffffff&fg=000000&s=0 "{[H,A] = HA - AH}")
L-infinity instability of Prandtl layers
In 1904, Prandtl introduced his famous boundary layer theory in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to 
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).
Stability of source defects in oscillatory media
Patterns are ubiquitous in nature, and understanding their formation and their dynamical behavior is always challenging and of great interest. Examples include patterns in fluids (e.g., Rayleigh-Benard convection between two flat plates, Taylor-Couette flow between rotating cylinders, surface waves in hydrothermal fluid flows,…), as well as in nonlinear optics, oscillatory chemical reactions and excitable biological media. Many of them arise from linear instabilities of an homogenous equilibrium, having space, time, or space-time periodic coherent structures such as wave trains (spatially periodic travelling waves). In presence of boundaries or defects, complex patterns form and thus break the symmetry or the periodic structures. Below, I shall briefly discuss some defect structures and my recent work on the subject.
Graduate Student Seminar: Topics in Fluid Dynamics
Today, I give a Graduate Student Seminar lecture whose goal is to introduce to the first and second year graduate students at Penn State a few topics of research in Fluid Dynamics. There are many recent exciting developments in the field, which I only have time to present a few (many students haven’t taken any PDE course!). You may find the slides of my lecture here (up to many details delivered on the board!). You may also enjoy my similar lecture on Kinetic Theory of Gases, also aiming at first and second year students.