Steady vortex patches on flat torus with a constant background vorticity
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910776942395392 |
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| author | Sakajo, Takashi Zou, Changjun |
| author_facet | Sakajo, Takashi Zou, Changjun |
| contents | We construct a series of vortex patch solutions in a doubly-periodic rectangular domain (flat torus), which is accomplished by studying the contour dynamic equation for patch boundaries. We will illustrate our key idea by discussing the single-layered patches as the most fundamental configuration, and then investigate the general construction for $N$ patches near a point vortex equilibrium. Different with the case of bounded domains in $\mathbb R^2$, a constant background vorticity will arise from the compact nature of flat torus, and the $2$-dimensional translational invariance will bring troubles on determining patch locations. To overcome these two difficulties, we will add additional terms for background vorticity and introduce a centralized condition for location vector. By utilizing the regularity difference of terms in contour dynamic equations, we also obtain the $C^\infty$ regularity and convexity of boundary curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Steady vortex patches on flat torus with a constant background vorticity Sakajo, Takashi Zou, Changjun Analysis of PDEs 76B07 We construct a series of vortex patch solutions in a doubly-periodic rectangular domain (flat torus), which is accomplished by studying the contour dynamic equation for patch boundaries. We will illustrate our key idea by discussing the single-layered patches as the most fundamental configuration, and then investigate the general construction for $N$ patches near a point vortex equilibrium. Different with the case of bounded domains in $\mathbb R^2$, a constant background vorticity will arise from the compact nature of flat torus, and the $2$-dimensional translational invariance will bring troubles on determining patch locations. To overcome these two difficulties, we will add additional terms for background vorticity and introduce a centralized condition for location vector. By utilizing the regularity difference of terms in contour dynamic equations, we also obtain the $C^\infty$ regularity and convexity of boundary curves. |
| title | Steady vortex patches on flat torus with a constant background vorticity |
| topic | Analysis of PDEs 76B07 |
| url | https://arxiv.org/abs/2501.04271 |