Global infinite energy solutions for the 2D gravity water waves system

TL;DR

This paper establishes global existence and modified scattering for 2D gravity water waves with initial data allowing infinite energy, removing previous momentum constraints and broadening the understanding of wave behavior in infinite depth.

Contribution

It proves global solutions and scattering for 2D water waves with minimal initial regularity, extending results to infinite energy scenarios without previous velocity assumptions.

Findings

Global existence of solutions for infinite energy initial data.

Modified scattering behavior of solutions.

Removal of previous momentum condition assumptions.

Abstract

We prove global existence and modified scattering property for the solutions of the $2D$ gravity water waves system in the infinite depth setting for a class of initial data, which is only required to be small above the level $\dot{H}^{1/5}\times \dot{H}^{1/5+1/2}$. No assumption is assumed below this level, therefore, it allows to have infinite energy. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu-Pusateri, Alazard-Delort and Ifrim-Tataru is removed.