The Landau equation is one of the fundamental models of statistical physics for describing the evolution of plasmas. Nevertheless, the existence of global-in-time smooth solutions remains widely open, even for the homogeneous equation. I will discuss some recent progress in constructing local-in-time strong solutions for large initial data under minimal integrability assumptions. Our method relies crucially on a priori short time smoothing estimates obtained via the De Giorgi method. While for Landau these smoothing estimates require some integrability assumptions on the initial data, these assumptions may be removed for critical perturbations of the Landau equation. Consequently, arbitrarily small critical perturbations of the Landau equation admit global-in-time smooth solutions for generic, possibly large initial data.

We will go over the main ideas used in showing that as long as super-critical Holder semi-norms of the classical solution to the incompressible Navier-Stokes system (in any dimension) are under control, the solution remains smooth. The key idea is exploiting an enhanced regularity effect coming from the transport term at the level of a simple one-dimensional drift-diffusion equation, allowing us to break the criticality barrier. We then employ ideas introduced by Kiselev, Nazarov, Volberg, and Shterenberg to propagate this to abstract drift-diffusion equations, providing to our knowledge the very first reasonable extension of the celebrated parabolic regularity result of Nash to an equation that is not in divergence form. Such an approach coupled with subtle pressure estimates due to Silvestre allows us to rigorously treat the incompressible Navier-Stokes as a perturbation of drift-diffusion, obtaining, to our knowledge, the very first genuinely super-critical regularity criterion for this system of equations. Zoom link: https://tamu.zoom.us/j/93413906718?pwd=VDlXYS8zbm9uQ2dMWGowcGVOeGRPdz09

In this talk, we will discuss dispersive and local decay estimates for a class of matrix Schrödinger operators that naturally arise from the linearization of focusing nonlinear Schrödinger equations around a solitary wave. We review the spectral properties of these linearized operators, and discuss how threshold resonances may appear in their spectrum. In the presence of threshold resonances, it will be shown that the slowdown of the local decay rate can be pinned down to a finite rank operator corresponding to the threshold resonances. Some applications for the linearized equation for the 1D focusing cubic Schrödinger equation will be discussed.