Long-term regularity of 2D gravity water waves

TL;DR

This paper proves that small initial data for 2D gravity water waves lead to solutions that remain regular for a longer time than previously known, with lifespan proportional to the inverse fourth power of the initial data size.

Contribution

It improves the known lifespan estimates for solutions of the 2D gravity water wave problem from  to , and extends analysis to periodic waves bridging non-periodic and periodic cases.

Findings

Solutions with small initial data are well-posed for at least
/ time.

The lifespan of solutions is extended beyond previous estimates.

A new analysis bridges the gap between non-periodic and periodic water waves.

Abstract

The two dimensional gravity water wave problem concerns the motion of an incompressible fluid occupying half the 2D space and flowing under its own gravity. In this paper we study long-term regularity of solutions evolving from small but non-localized initial data. Our main result is that if the $H^s$ norm of the initial data is $\epsilon$, where $s \gg 1$ and $\epsilon \ll 1$, then the equation is wellposed at least for a time proportional to $\epsilon^{-4}$, improving on the $\epsilon^{-3}$ lifespan obtained in \cite{BeFePu,Wu2DL}. We also study period water waves and show a lifespan bridging the gap between non-periodic waves and waves with a period of 1.