We consider a family of quasiperiodic solutions of the nonlinear Schrodinger equation on the 2-torus, namely the family of finite-gap solutions (tori). These solutions are inherited by the 2D equation from its completely integrable 1D counterpart (NLS on the circle) by considering solutions that only depend on one variable. Despite being linearly stable, we prove that these tori (under some genericness conditions) are nonlinearly unstable in the following strong sense: there exists solutions that start very close to those tori in certain Sobolev spaces, but eventually become larger than any given factor at later times. This is the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed. (joint work with M. Guardia (UPC, Barcelona), E. Haus (University of Naples), M. Procesi (Roma Tre), and A. Maspero (SISSA))

 In an exciting paper, J. Bedrossian and N. Masmoudi established the stability of the 2D Couette flow in Gevrey spaces of index greater than 1/2. I will talk about recent joint work with N. Masmoudi, which proves, in the opposite direction, the instability of the Couette flow in Gevrey spaces of index smaller than 1/2. This confirms, to a large extent, that the transient growth predicted heuristically in earlier works does exist and has the expected strength. The proof is based on the fremawork of the stability result, with a few crucial new observations. I will also discuss related works regarding Landau damping, and possible extensions to infinite time.

This talk concerns a PDE system that models tumor growth. We show that a novel free boundary problem arises via the incompressible limit of this model. We take a viscosity solutions approach; however, since the system lacks maximum principle, there are interesting challenges to overcome. This is joint work with Inwon Kim.

Given a planar infinity harmonic function u, for each
$\alpha>0$ we show a quantitative $W^{1,\,2}_{\loc}$-estimate of
$|Du|^{\alpha}$, which is sharp when $\alpha\to 0$.  As a consequence we
obtain an $L^p$-Liouville property for infinity harmonic functions in
the whole plane
 

We consider the nonlinear Schroedinger equation posed on a large box of characteristic size $L$, and ask about its effective dynamics for very long time scales. After pointing out some “more or less” trivial time scales along which the effective dynamics can be easily described, we start inspecting some much longer time scales where we notice some non-trivial dynamical behaviors. Particularly, the end goal of such an analysis is to reach the so-called “kinetic time scale”, at which it is conjectured that the effective dynamics is governed by a kinetic equation called the “wave kinetic equation”. This is the subject of wave turbulence theory. We will discuss some recent advances towards this end goal. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah.