A pseudo-local property of gravity water waves system

TL;DR

This paper demonstrates that gravity water waves exhibit a pseudo-local property, meaning distant regions have minimal influence on each other over time, supported by new estimates and a paradifferential calculus framework.

Contribution

It introduces a weighted contraction estimate in uniformly local Sobolev spaces, revealing the pseudo-local nature of gravity water waves and establishing a new spatial decay property.

Findings

Distant regions have a small effect on local wave dynamics

Established a paradifferential calculus in weighted Sobolev spaces

Proved a new spatial decay property of water waves

Abstract

By proving a weighted contraction estimate in uniformly local Sobolev spaces for the flow of gravity water waves, we show that this nonlocal system is in fact pseudo-local in the following sense: locally in time, the dynamic far away from a given bounded region has a small effect on that region (again, in a sense that we will make precise in the article). Our estimate on the flow also implies a new spatial decay property of the waves. To prove this result, we establish a paradifferential calculus theory in uniformly local Sobolev spaces with weights.