Analytical Study of a generalised Dirichlet-Neumann operator and application to three-dimensional water waves on Beltrami flows
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929372178415616 |
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| author | Groves, Mark D. Nilsson, Dag Pasquali, Stefano Wahlén, Erik |
| author_facet | Groves, Mark D. Nilsson, Dag Pasquali, Stefano Wahlén, Erik |
| contents | In this paper we consider three-dimensional 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 discuss a recent variational formulation of the problem 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 rigorously prove existence of doubly periodic gravity-capillary steady waves and construct approximate doubly periodic gravity steady waves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01702 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Analytical Study of a generalised Dirichlet-Neumann operator and application to three-dimensional water waves on Beltrami flows Groves, Mark D. Nilsson, Dag Pasquali, Stefano Wahlén, Erik Analysis of PDEs 76B15, 76B45, 47G30 In this paper we consider three-dimensional 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 discuss a recent variational formulation of the problem 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 rigorously prove existence of doubly periodic gravity-capillary steady waves and construct approximate doubly periodic gravity steady waves. |
| title | Analytical Study of a generalised Dirichlet-Neumann operator and application to three-dimensional water waves on Beltrami flows |
| topic | Analysis of PDEs 76B15, 76B45, 47G30 |
| url | https://arxiv.org/abs/2307.01702 |