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Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
About this Title
Thomas Alazard, DMA, École normale supérieure et CNRS UMR 8553, 45 rue dõUlm, 75005 Paris, France, Nicolas Burq, Univiversité Paris-Sud, Département de Mathématiques, 91405 Orsay, France and Claude Zuily, Univiversité Paris-Sud, Département de Mathématiques, 91405 Orsay, France
Publication: Memoirs of the American Mathematical Society
Publication Year:
2018; Volume 256, Number 1229
ISBNs: 978-1-4704-3203-4 (print); 978-1-4704-4921-6 (online)
DOI: https://doi.org/10.1090/memo/1229
Published electronically: August 16, 2018
Keywords: Water waves,
Strichartz estimates,
paradifferential calculus
MSC: Primary 35Q35, 35S50, 35S15, 76B15
Table of Contents
Chapters
- 1. Introduction
- 2. Strichartz estimates
- 3. Cauchy problem
- A. Paradifferential calculus
- B. Tame estimates for the Dirichlet-Neumann operator
- C. Estimates for the Taylor coefficient
- D. Sobolev estimates
- E. Proof of a technical result
Abstract
This memoir is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves, we can consider solutions such that the curvature of the initial free surface does not belong to $L^2$.
The proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and Hölder estimates. We first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then we use the dispersive properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.
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