Almost global wellposedness of the 2-D full water wave problem

We consider the problem of global in time existence and uniqueness of solutions of the 2-D infinite depth full water wave equation. It is known that this equation has a solution for a time period 0,T/ε] for initial data of the form ε Ψ, where T depends only on Ψ. In this paper, we show that for such data there exists a unique solution for a time period 0,e ^T/ε ]. This is achieved by better understandings of the nature of the nonlinearity of the full water wave equation. Financial support provided in part by NSF grant DMS-0400643.