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Quasi-Periodic Standing Wave Solutions of Gravity-Capillary Water Waves
About this Title
Massimiliano Berti, Massimiliano Berti, SISSA, Via Bonomea 265, 34136, Trieste, Italy and Riccardo Montalto, Riccardo Montalto, University of Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 263, Number 1273
ISBNs: 978-1-4704-4069-5 (print); 978-1-4704-5654-2 (online)
DOI: https://doi.org/10.1090/memo/1273
Published electronically: February 10, 2020
Keywords: KAM for PDEs,
Water waves,
quasi-periodic solutions,
standing waves
MSC: Primary 76B15, 37K55, 76D45; Secondary 37K50, 35S05
Table of Contents
Chapters
- 1. Introduction and main result
- 2. Functional setting
- 3. Transversality properties of degenerate KAM theory
- 4. Nash-Moser theorem and measure estimates
- 5. Approximate inverse
- 6. The linearized operator in the normal directions
- 7. Almost diagonalization and invertibility of ${\mathcal L}_\omega$
- 8. The Nash-Moser iteration
- A. Tame estimates for the flow of pseudo-PDEs
Abstract
We prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable $x$) of a $2$-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.- Thomas Alazard and Pietro Baldi, Gravity capillary standing water waves, Arch. Ration. Mech. Anal. 217 (2015), no. 3, 741–830. MR 3356988, DOI 10.1007/s00205-015-0842-5
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