Cauchy theory for the gravity water waves system with non localized initial data

TL;DR

This paper establishes local well-posedness for the gravity water waves system with non-decaying initial data using $L^2$-based Sobolev spaces, extending classical results to rough, non-localized initial conditions.

Contribution

It develops the first local Cauchy theory for water waves with non-localized data in Kato's Sobolev spaces, including well-posedness in H"older spaces and addressing Boussinesq's question.

Findings

Proves well-posedness without loss of derivatives for rough initial data.

Extends classical theory to non-decaying initial conditions.

Solves Boussinesq's water waves problem in a canal setting.

Abstract

In this article, we develop the local Cauchy theory for the gravity water waves system, for rough initial data which do not decay at infinity. We work in the context of $L^2$-based uniformly local Sobolev spaces introduced by Kato. We prove a classical well-posedness result (without loss of derivatives). Our result implies also a local well-posedness result in H\"older spaces (with loss of $d/2$ derivatives). As an illustration, we solve a question raised by Boussinesq on the water waves problem in a canal. We take benefit of an elementary observation to show that the strategy suggested by Boussinesq does indeed apply to this setting.