## Special Colloquium

Zaher Hani

Courant Institute, NYU

Out-of-equilibrium dynamics of nonlinear dispersive equations: From energy cascades to weak turbulence

**Abstract:**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 mathematical problems involved in studying them. Afterwards, we will describe two main approaches that were adopted (in the mathematics and physics communities) 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.

Monday December 2, 2013 at 3:00 PM in SEO 636