Cauchy Theory for the Water Waves System in an Analytic Framework

Abstract

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 $\epsilon$ in a space of analytic functions which have a holomorphic extension in a strip of size $\sigma$, then the solution exists up to a time of size $C/\epsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma -C^{\prime }\epsilon t$.