Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917867840077824 |
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| author | Fan, Boquan Wang, Yuchen Zhan, Weicheng |
| author_facet | Fan, Boquan Wang, Yuchen Zhan, Weicheng |
| contents | We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler equation in the unit disc, both in the smooth setting and the patch setting. In the patch setting, we prove that every uniformly rotating patch with angular velocity $Ω\le 0$ or $Ω\ge 1/2$ must be radial, where both bounds are sharp. The conclusion holds under the assumption that the rotating patch considered is disconnected, with its boundaries consisting of several Jordan curves. We also show that every uniformly rotating smooth solution $ω_0$ must be radially symmetric if its angular velocity $Ω\le \inf ω_0/2$ or $Ω\ge \sup ω_0/2$. The proof is based on the symmetry properties of non-negative solutions to elliptic problems. A newly tailored approach is developed to address the symmetries of non-negative solutions to piecewise coupled semi-linear elliptic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05973 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc Fan, Boquan Wang, Yuchen Zhan, Weicheng Analysis of PDEs We study the radial symmetry properties of stationary and uniformly rotating solutions of the 2D Euler equation in the unit disc, both in the smooth setting and the patch setting. In the patch setting, we prove that every uniformly rotating patch with angular velocity $Ω\le 0$ or $Ω\ge 1/2$ must be radial, where both bounds are sharp. The conclusion holds under the assumption that the rotating patch considered is disconnected, with its boundaries consisting of several Jordan curves. We also show that every uniformly rotating smooth solution $ω_0$ must be radially symmetric if its angular velocity $Ω\le \inf ω_0/2$ or $Ω\ge \sup ω_0/2$. The proof is based on the symmetry properties of non-negative solutions to elliptic problems. A newly tailored approach is developed to address the symmetries of non-negative solutions to piecewise coupled semi-linear elliptic equations. |
| title | Radial symmetry of stationary and uniformly-rotating solutions to the 2D Euler equation in a disc |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.05973 |