2019/10/11 by Albert Ai, Ai, Albert, Mihaela Ifrim +3
Earth and Planetary Sciences · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing
paper · pdf · doi:10.48550/arxiv.1910.05323
openalex publication_date 2019/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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].