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Analytical study of a generalised Dirichlet–Neumann operator and application to three-dimensional water waves on Beltrami flows

Groves, M. D. ; Nilsson, D. LU ; Pasquali, S. LU and Wahlén, E. LU orcid (2024) In Journal of Differential Equations 413. p.129-189
Abstract

We consider three-dimensional doubly periodic steady water waves with vorticity, under the action of gravity and surface tension; in particular we consider so-called Beltrami flows, for which the velocity field and the vorticity are collinear. We adapt a recent formulation of the corresponding problem for localised waves which involves a generalisation of the classical Dirichlet–Neumann operator. We study this operator in detail, extending some well-known results for the classical Dirichlet–Neumann operator, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and the asymptotic expansion of its associated symbol. A new formulation of the problem as a single equation for the wave... (More)

We consider three-dimensional doubly periodic steady water waves with vorticity, under the action of gravity and surface tension; in particular we consider so-called Beltrami flows, for which the velocity field and the vorticity are collinear. We adapt a recent formulation of the corresponding problem for localised waves which involves a generalisation of the classical Dirichlet–Neumann operator. We study this operator in detail, extending some well-known results for the classical Dirichlet–Neumann operator, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and the asymptotic expansion of its associated symbol. A new formulation of the problem as a single equation for the wave profile is also presented and discussed in a similar vein. As an application of these results we prove existence of doubly periodic gravity-capillary steady waves and construct approximate doubly periodic gravity steady waves.

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author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Beltrami flows, Vorticity, Water waves
in
Journal of Differential Equations
volume
413
pages
61 pages
publisher
Academic Press
external identifiers
  • scopus:85202736839
ISSN
0022-0396
DOI
10.1016/j.jde.2024.08.039
language
English
LU publication?
yes
id
a3912e0e-e044-456a-a922-c2eb8dfddccd
date added to LUP
2024-11-12 16:50:53
date last changed
2025-04-04 14:48:11
@article{a3912e0e-e044-456a-a922-c2eb8dfddccd,
  abstract     = {{<p>We consider three-dimensional doubly periodic steady water waves with vorticity, under the action of gravity and surface tension; in particular we consider so-called Beltrami flows, for which the velocity field and the vorticity are collinear. We adapt a recent formulation of the corresponding problem for localised waves which involves a generalisation of the classical Dirichlet–Neumann operator. We study this operator in detail, extending some well-known results for the classical Dirichlet–Neumann operator, such as the Taylor expansion in homogeneous powers of the wave profile, the computation of its differential and the asymptotic expansion of its associated symbol. A new formulation of the problem as a single equation for the wave profile is also presented and discussed in a similar vein. As an application of these results we prove existence of doubly periodic gravity-capillary steady waves and construct approximate doubly periodic gravity steady waves.</p>}},
  author       = {{Groves, M. D. and Nilsson, D. and Pasquali, S. and Wahlén, E.}},
  issn         = {{0022-0396}},
  keywords     = {{Beltrami flows; Vorticity; Water waves}},
  language     = {{eng}},
  pages        = {{129--189}},
  publisher    = {{Academic Press}},
  series       = {{Journal of Differential Equations}},
  title        = {{Analytical study of a generalised Dirichlet–Neumann operator and application to three-dimensional water waves on Beltrami flows}},
  url          = {{http://dx.doi.org/10.1016/j.jde.2024.08.039}},
  doi          = {{10.1016/j.jde.2024.08.039}},
  volume       = {{413}},
  year         = {{2024}},
}