Speaker: Mihaela Ifrim (UC Berkeley)

Title: Long time solutions for two dimensional water waves

Abstract: This is joint work with Daniel Tataru, and in parts with John Hunter. My talk is concerned with the infinite depth water wave equation in two space dimensions, with either gravity or surface tension. Both cases will be discussed in parallel. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. For the gravity water waves there are several results available; they have been recently obtained by Wu, Alazard-Burq-Zuily and Ionescu-Pusateri using different coordinates and methods. In the capillary water waves case, we were the first to establish a global result. Our goal is improve the understanding of these problems by providing a single setting for both cases, and presenting simpler proofs. The talk will be as self contained as the time permits.