# A Dispersive Estimate for the Linearized Water-Waves Equations in Finite Depth

A Dispersive Estimate for the Linearized Water-Waves Equations in Finite Depth We prove a dispersive estimate for the solutions of the linearized Water-Waves equations in dimension $${d=1}$$ d = 1 and $${d=2}$$ d = 2 in presence of a flat bottom. Adapting the proof from Aynur (An optimal decay estimate for the linearized water wave equation in 2d. arXiv:1411.0963 , 2014) in the case of infinite depth, we prove a decay with respect to time t of order $${\vert t \vert^{-d/3}}$$ | t | - d / 3 for solutions with initial data $${\varphi}$$ φ such that $${\vert\varphi\vert_{H^1}}$$ | φ | H 1 , $${\vert x\varphi\vert_{H^1}}$$ | x φ | H 1 are bounded. We also give variants to this result with different decays for a more convenient use of the dispersive estimate. We then give an existence result for the full Water-Waves equations in weighted spaces for practical uses of the proven dispersive estimates. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# A Dispersive Estimate for the Linearized Water-Waves Equations in Finite Depth

, Volume 19 (3) – Sep 2, 2016
32 pages

/lp/springer_journal/a-dispersive-estimate-for-the-linearized-water-waves-equations-in-p062paH3LN
Publisher
Springer International Publishing
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-016-0286-1
Publisher site
See Article on Publisher Site

### Abstract

We prove a dispersive estimate for the solutions of the linearized Water-Waves equations in dimension $${d=1}$$ d = 1 and $${d=2}$$ d = 2 in presence of a flat bottom. Adapting the proof from Aynur (An optimal decay estimate for the linearized water wave equation in 2d. arXiv:1411.0963 , 2014) in the case of infinite depth, we prove a decay with respect to time t of order $${\vert t \vert^{-d/3}}$$ | t | - d / 3 for solutions with initial data $${\varphi}$$ φ such that $${\vert\varphi\vert_{H^1}}$$ | φ | H 1 , $${\vert x\varphi\vert_{H^1}}$$ | x φ | H 1 are bounded. We also give variants to this result with different decays for a more convenient use of the dispersive estimate. We then give an existence result for the full Water-Waves equations in weighted spaces for practical uses of the proven dispersive estimates.

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Sep 2, 2016

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