Periodic nonlinear Schrödinger equations and the evolution of its energy spectrum
Periodic nonlinear Schrödinger equations and the evolution of its energy spectrum
In-Person and Livestream: TBD
In this Ìýcourse we Ìýwill investigate some questions Ìýrelated to weak turbulence theory by using as explicit example of wave interactions the solutions to the 2D periodic cubic ÌýSchrödinger equation. We will start by recalling ÌýStrichartz estimates and well-posedness. ÌýWe will then explain how the evolution of theÌýenergy spectrum related to this equation can be studied inÌýÌýtwo different ways. We will first work on a method proposed by Bourgain and involving the growth of high Sobolev norms. ÌýThen we will present some recent results on energy cascade via the analysis of radial Ìýsolutions to ÌýtheÌýwave kinetic equation, Ìýwhich is often used as the effective equation for the energy spectrum mentioned above. Ìý