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Symmetric doubly periodic gravity-capillary waves with small vorticity

Svensson Seth, Douglas ; Varholm, Kristoffer and Wahlén, Erik LU (2022)
Abstract
We construct small amplitude gravity-capillary water waves with small nonzero vorticity, in three spatial dimensions, bifurcating from uniform flows. The waves are symmetric, and periodic in both horizontal coordinates. The proof is inspired by Lortz' construction of magnetohydrostatic equilibria in reflection-symmetric toroidal domains. It relies on a representation of the vorticity as the cross product of two gradients, and on prescribing a functional relationship between the Bernoulli function and the orbital period of the water particles. The presence of the free surface introduces significant new challenges. In particular, the resulting free boundary problem is not elliptic, and the involved maps incur a loss of regularity under... (More)
We construct small amplitude gravity-capillary water waves with small nonzero vorticity, in three spatial dimensions, bifurcating from uniform flows. The waves are symmetric, and periodic in both horizontal coordinates. The proof is inspired by Lortz' construction of magnetohydrostatic equilibria in reflection-symmetric toroidal domains. It relies on a representation of the vorticity as the cross product of two gradients, and on prescribing a functional relationship between the Bernoulli function and the orbital period of the water particles. The presence of the free surface introduces significant new challenges. In particular, the resulting free boundary problem is not elliptic, and the involved maps incur a loss of regularity under Fréchet differentiation. Nevertheless, we show that a version of the Crandall--Rabinowitz local bifurcation method applies, by carefully tracking the loss of regularity. (Less)
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author
; and
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publishing date
type
Working paper/Preprint
publication status
published
subject
publisher
arXiv.org
DOI
10.48550/arXiv.2204.13093
language
English
LU publication?
yes
id
6716f545-2e07-4d2d-bc20-92d18060ecc2
date added to LUP
2023-03-25 18:56:06
date last changed
2023-04-27 10:58:05
@misc{6716f545-2e07-4d2d-bc20-92d18060ecc2,
  abstract     = {{We construct small amplitude gravity-capillary water waves with small nonzero vorticity, in three spatial dimensions, bifurcating from uniform flows. The waves are symmetric, and periodic in both horizontal coordinates. The proof is inspired by Lortz' construction of magnetohydrostatic equilibria in reflection-symmetric toroidal domains. It relies on a representation of the vorticity as the cross product of two gradients, and on prescribing a functional relationship between the Bernoulli function and the orbital period of the water particles. The presence of the free surface introduces significant new challenges. In particular, the resulting free boundary problem is not elliptic, and the involved maps incur a loss of regularity under Fréchet differentiation. Nevertheless, we show that a version of the Crandall--Rabinowitz local bifurcation method applies, by carefully tracking the loss of regularity.}},
  author       = {{Svensson Seth, Douglas and Varholm, Kristoffer and Wahlén, Erik}},
  language     = {{eng}},
  note         = {{Preprint}},
  publisher    = {{arXiv.org}},
  title        = {{Symmetric doubly periodic gravity-capillary waves with small vorticity}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2204.13093}},
  doi          = {{10.48550/arXiv.2204.13093}},
  year         = {{2022}},
}