Sharp Strichartz estimates for water waves systems

TL;DR

This paper establishes optimal dispersive Strichartz estimates for nonlinear water wave systems, including gravity and gravity-capillary waves, in both 2D and 3D, advancing understanding of their regularity properties.

Contribution

It provides the first proof of optimal Strichartz estimates for fully nonlinear water wave systems, combining paradifferential calculus with localized dispersive estimates.

Findings

Optimal Strichartz estimates for gravity waves

Semi-classical estimates for gravity-capillary waves

Applicable to both 2D and 3D water wave models

Abstract

Water waves are well-known to be dispersive at the linearization level. Considering the fully nonlinear systems, we prove for reasonably smooth solutions the optimal Strichartz estimates for pure gravity waves and the semi-classical Strichartz estimates for gravity-capillary waves; for both 2D and 3D waves. Here, by optimal we mean the gains of regularity (over the Sobolev embedding from Sobolev spaces to H\"older spaces) obtained for the linearized systems. Our proofs combine the paradifferential reductions of Alazard-Burq-Zuily with a dispersive estimate using a localized wave package type parametrix of Koch-Tataru.