Mathematics Colloquium: Zaher Hani (Courant Institute), 'Energy cascades and weak turbulence for nonlinear dispersive equations'

Event Date: 

Wednesday, January 8, 2014 -
3:30pm to 4:30pm

Event Location: 

  • 6635 South Hall

Out-of-equilibrium dynamics are a characteristic feature of the long-time behavior of nonlinear dispersive equations (PDEs) on bounded domains. Energy cascades and weak (or wave) turbulence are main aspects of such out-of-equilibrium dynamics. In this talk, we will start by explaining what all those concepts mean, why they arise, and the analytical problems involved in studying them. Afterwards, we will describe two main approaches that were adopted to capture these phenomena. The first approach is based on a relation between energy cascades and the growth of Sobolev norms of solutions. The second is based on deriving "effective equations" for the dynamics by taking various limits of the original system (this is the guiding philosophy of weak turbulence theory). We will report on some recent progress on both approaches.