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LARGE-AMPLITUDE STEADY GRAVITY WATER WAVES WITH GENERAL VORTICITY AND CRITICAL LAYERS

Wahlén, Erik LU orcid and Weber, Jörg LU (2024) In Duke Mathematical Journal 173(11). p.2197-2258
Abstract

We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging, and no assumptions regarding the absence of stagnation points and critical layers are made. Using conformal mappings and a new reformulation of Bernoulli’s equation, we uncover an equivalent formulation as “identity plus compact,” which is amenable to Rabinowitz’s global bifurcation theorem. This allows us to construct a global connected set of solutions, bifurcating from laminar flows with a flat surface. Moreover, a nodal analysis is carried out for these solutions under a certain spectral assumption involving the vorticity function. Lastly,... (More)

We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging, and no assumptions regarding the absence of stagnation points and critical layers are made. Using conformal mappings and a new reformulation of Bernoulli’s equation, we uncover an equivalent formulation as “identity plus compact,” which is amenable to Rabinowitz’s global bifurcation theorem. This allows us to construct a global connected set of solutions, bifurcating from laminar flows with a flat surface. Moreover, a nodal analysis is carried out for these solutions under a certain spectral assumption involving the vorticity function. Lastly, downstream waves are investigated in more detail.

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author
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type
Contribution to journal
publication status
published
subject
in
Duke Mathematical Journal
volume
173
issue
11
pages
62 pages
publisher
Duke University Press
external identifiers
  • scopus:85200454460
ISSN
0012-7094
DOI
10.1215/00127094-2023-0054
language
English
LU publication?
yes
id
6bf2c2e7-cb07-4d4e-81a9-bbb3e21a15d3
date added to LUP
2024-11-05 15:29:58
date last changed
2024-11-05 15:39:40
@article{6bf2c2e7-cb07-4d4e-81a9-bbb3e21a15d3,
  abstract     = {{<p>We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging, and no assumptions regarding the absence of stagnation points and critical layers are made. Using conformal mappings and a new reformulation of Bernoulli’s equation, we uncover an equivalent formulation as “identity plus compact,” which is amenable to Rabinowitz’s global bifurcation theorem. This allows us to construct a global connected set of solutions, bifurcating from laminar flows with a flat surface. Moreover, a nodal analysis is carried out for these solutions under a certain spectral assumption involving the vorticity function. Lastly, downstream waves are investigated in more detail.</p>}},
  author       = {{Wahlén, Erik and Weber, Jörg}},
  issn         = {{0012-7094}},
  language     = {{eng}},
  number       = {{11}},
  pages        = {{2197--2258}},
  publisher    = {{Duke University Press}},
  series       = {{Duke Mathematical Journal}},
  title        = {{LARGE-AMPLITUDE STEADY GRAVITY WATER WAVES WITH GENERAL VORTICITY AND CRITICAL LAYERS}},
  url          = {{http://dx.doi.org/10.1215/00127094-2023-0054}},
  doi          = {{10.1215/00127094-2023-0054}},
  volume       = {{173}},
  year         = {{2024}},
}