Gravity water waves over constant vorticity flows: From laminar flows to touching waves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913920586874880 |
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| author | Gonçalves, Francisco |
| author_facet | Gonçalves, Francisco |
| contents | In a recent paper, Hur & Wheeler [J. Differential Equations, 338:572-590, 2022] proved the existence of periodic steady water waves over an infinitely deep, two-dimensional and constant vorticity flow under the influence of gravity. These solutions include overhanging wave profiles, some of which exhibit surfaces that touch at a point and thereby enclose a bubble of air. We extend these results by formulating a problem that encompasses both infinitely deep and finitely deep flows, and by proving the existence of a continuous curve of water waves that connects a laminar flow to a touching wave for fixed, nonzero gravity. This implies the existence of a wave profile featuring a vertical tangent at a point, which is not overhanging, and is referred to as a breaking wave. We also study the behaviour of critical layers, which are points where the horizontal velocity vanishes, near the surface. In particular, this result holds for arbitrary vorticity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_00417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gravity water waves over constant vorticity flows: From laminar flows to touching waves Gonçalves, Francisco Analysis of PDEs 35Q31, 35Q35, 76B03, 76B15 In a recent paper, Hur & Wheeler [J. Differential Equations, 338:572-590, 2022] proved the existence of periodic steady water waves over an infinitely deep, two-dimensional and constant vorticity flow under the influence of gravity. These solutions include overhanging wave profiles, some of which exhibit surfaces that touch at a point and thereby enclose a bubble of air. We extend these results by formulating a problem that encompasses both infinitely deep and finitely deep flows, and by proving the existence of a continuous curve of water waves that connects a laminar flow to a touching wave for fixed, nonzero gravity. This implies the existence of a wave profile featuring a vertical tangent at a point, which is not overhanging, and is referred to as a breaking wave. We also study the behaviour of critical layers, which are points where the horizontal velocity vanishes, near the surface. In particular, this result holds for arbitrary vorticity. |
| title | Gravity water waves over constant vorticity flows: From laminar flows to touching waves |
| topic | Analysis of PDEs 35Q31, 35Q35, 76B03, 76B15 |
| url | https://arxiv.org/abs/2505.00417 |