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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.16671 |
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| _version_ | 1866914928960471040 |
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| author | Hassainia, Zineb Hmidi, Taoufik Roulley, Emeric |
| author_facet | Hassainia, Zineb Hmidi, Taoufik Roulley, Emeric |
| contents | We examine the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic motion. We show that for the single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. The proof uses Nash-Moser scheme and KAM techniques, in the spirit of the recent work of Hassainia-Hmidi-Masmoudi on the leapfrogging motion, combined with complex geometry tools. Additionally, we employ a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices. This approach can be, for instance, applied to desingularize the motion of two symmetric dipoles (with four vortices) in a disc or a rectangle. To our knowledge, this is the first result showing the existence of non-rigid time-periodic motion for Euler equations in generic simply-connected bounded domain. This answers an open problem that has been pointed in the literature, for example by Bartsch-Sacchet. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_16671 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Desingularization of time-periodic vortex motion in bounded domains via KAM tools Hassainia, Zineb Hmidi, Taoufik Roulley, Emeric Analysis of PDEs We examine the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic motion. We show that for the single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of these non-degeneracy conditions, which are satisfied by a broad class of domains, including convex ones. The proof uses Nash-Moser scheme and KAM techniques, in the spirit of the recent work of Hassainia-Hmidi-Masmoudi on the leapfrogging motion, combined with complex geometry tools. Additionally, we employ a vortex duplication mechanism to generate synchronized time-periodic motion of multiple vortices. This approach can be, for instance, applied to desingularize the motion of two symmetric dipoles (with four vortices) in a disc or a rectangle. To our knowledge, this is the first result showing the existence of non-rigid time-periodic motion for Euler equations in generic simply-connected bounded domain. This answers an open problem that has been pointed in the literature, for example by Bartsch-Sacchet. |
| title | Desingularization of time-periodic vortex motion in bounded domains via KAM tools |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.16671 |