Two dimensional gravity waves at low regularity I: Energy estimates

TL;DR

This paper develops sharp cubic energy estimates for 2D water wave equations at low regularity, improving the regularity threshold for local well-posedness without relying on Strichartz estimates.

Contribution

Introduction of balanced cubic energy estimates that enhance previous cubic estimates while maintaining scale invariance and holomorphic coordinate formulation.

Findings

Lowered Sobolev regularity threshold for well-posedness.

Established new balanced cubic energy estimates.

Improved upon earlier energy estimates without Strichartz estimates.

Abstract

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and develop the techniques to prove a new class of energy estimates, which we call \emph{balanced cubic estimates}. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru [15], while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using any Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness, drastically improving earlier results obtained by Alazard-Burq-Zuily [5, 6], Hunter-Ifrim-Tataru [15] and Ai [2].