Cauchy theory for the water waves system in an analytic framework

TL;DR

This paper establishes local well-posedness for the gravity water waves system in an analytic framework, showing solutions persist for times inversely proportional to initial data size, with analyticity strip shrinking over time.

Contribution

It introduces a novel analytic approach to the water waves problem, proving existence and lifespan estimates in spaces of analytic functions with shrinking holomorphic strips.

Findings

Solutions exist up to time C/ε for initial data of size ε

The analyticity strip decreases linearly with time

Provides a rigorous framework for analytic water waves analysis

Abstract

In this paper we consider the Cauchy problem for gravity water waves, in a domain with a flat bottom and in arbitrary space dimension. We prove that if the data are of size $\varepsilon$ 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 $C/\varepsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $\sigma - C'\varepsilon t$.