Cauchy Theory for the Water Waves System in an Analytic Framework

In this paper we consider the Cauchy problem for the gravity water-wave equations, in a domain with flat bottom and in arbitrary space dimension. We prove that if the data are of size ε in a space of analytic functions which have a holomorphic extension in a strip of size σ, then the solution exists up to a time of size C/ε in a space of analytic functions having at time t a holomorphic extension in a strip of size σ−C′εt.