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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2311.15413 |
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| _version_ | 1866914680992169984 |
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| author | Gancedo, Francisco Hidalgo-Torné, Antonio |
| author_facet | Gancedo, Francisco Hidalgo-Torné, Antonio |
| contents | This paper studies the Cauchy problem for a helical vortex filament evolving by the 3D incompressible Navier-Stokes equations. We prove global-in-time well-posedness and smoothing of solutions with initial vorticity concentrated on a helix. We provide a local-in-time well-posedness result for vortex filaments periodic in one spatial direction, and show that solutions with helical initial data preserve this symmetry. We follow the approach of [4], where the analogue local-in-time result has been obtained for closed vortex filaments in $\mathbb{R}^3$. Next, we apply local energy weak solutions theory with a novel estimate for helical functions in non-helical domains to uniquely extend the solutions globally in time. This is the first global-in-time well-posedness result for a vortex filament without size restriction and without vanishing swirl assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15413 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Cauchy problem for 3D Navier-Stokes helical vortex filament Gancedo, Francisco Hidalgo-Torné, Antonio Analysis of PDEs 76D05 This paper studies the Cauchy problem for a helical vortex filament evolving by the 3D incompressible Navier-Stokes equations. We prove global-in-time well-posedness and smoothing of solutions with initial vorticity concentrated on a helix. We provide a local-in-time well-posedness result for vortex filaments periodic in one spatial direction, and show that solutions with helical initial data preserve this symmetry. We follow the approach of [4], where the analogue local-in-time result has been obtained for closed vortex filaments in $\mathbb{R}^3$. Next, we apply local energy weak solutions theory with a novel estimate for helical functions in non-helical domains to uniquely extend the solutions globally in time. This is the first global-in-time well-posedness result for a vortex filament without size restriction and without vanishing swirl assumptions. |
| title | On the Cauchy problem for 3D Navier-Stokes helical vortex filament |
| topic | Analysis of PDEs 76D05 |
| url | https://arxiv.org/abs/2311.15413 |