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.
I consider solutions with asymptotic self-similarity. The behavior shows an invariance which comes naturally from nonlinearity. The basic model is Lane-Emden equation. Solution structures depend on the dimension as well as the exponent describing the nonlinearity. More generally, I will explain the corresponding result for quasilinear equations in the radial setting.
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.
In this talk, we will show that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., 229 (2005), no. 1, 62–94] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.