We discuss some recent work on stability of solutions to linear and nonlinear wave equations on asymptotically flat space times. This is joint work with Jason Metcalfe, Jesus Oliver, and Daniel Tataru.
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.
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.