Sharp Strichartz estimates for water waves systems

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 Hölder 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.

Some typos fixed

References in corpus (2)

• Two dimensional water waves in holomorphic coordinates
• Strichartz estimates and local existence for the capillary water waves with non-Lipschitz initial velocity

Cited by in corpus (2)

• Long-term regularity of 3D gravity water waves
• Low regularity solutions for gravity water waves II: The 2D case