Two dimensional gravity waves at low regularity I: Energy 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].
Cite
@article{arxiv.1910.05323,
title = {Two dimensional gravity waves at low regularity I: Energy estimates},
author = {Albert Ai and Mihaela Ifrim and Daniel Tataru},
journal= {arXiv preprint arXiv:1910.05323},
year = {2023}
}
Comments
63 pages. The current version introduces substantial revisions and changes in notation. The primary reason for the revision is to harmonize this paper with the second installment in the series