Abstract

A class of water wave problems concerns the dynamics of the free interface separating an inviscid, incompressible and irrotational fluid (water), under the influence of gravity, from a zero-density region (air). In these lectures I will present some recent methods and ideas developed concerning the local and global wellposedness of these problems, where the fluid region has either no fixed boundary, or has a fixed vertical wall. The emphasis will be on the understanding of the mathematical structure that leads to the results on global existence of small and smooth waves, local existence of arbitrary smooth waves, the persistence of water waves with angled crests, and in the case where there is a vertical wall, the interaction of the free interface with the wall. These lectures will be accessible to graduate students and post-doctors.