Desingularization of vortices for the incompressible Euler equation on a sphere
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arXiv
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| Format: | Preprint |
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2024
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| author | Cao, Daomin Li, Shuanglong Wang, Guodong |
| author_facet | Cao, Daomin Li, Shuanglong Wang, Guodong |
| contents | In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the polar axis. More importantly, these solutions ``converges" to a pair of point vortices with equal strength and opposite signs. The construction is achieved by maximizing the energy-impulse functional relative to a family of suitable rearrangement classes and analyzing the asymptotic behavior of the maximizers. Based on their variational characterization, we also prove the stability of these rotating solutions with respect to odd-symmetric perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07645 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Desingularization of vortices for the incompressible Euler equation on a sphere Cao, Daomin Li, Shuanglong Wang, Guodong Analysis of PDEs In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the polar axis. More importantly, these solutions ``converges" to a pair of point vortices with equal strength and opposite signs. The construction is achieved by maximizing the energy-impulse functional relative to a family of suitable rearrangement classes and analyzing the asymptotic behavior of the maximizers. Based on their variational characterization, we also prove the stability of these rotating solutions with respect to odd-symmetric perturbations. |
| title | Desingularization of vortices for the incompressible Euler equation on a sphere |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.07645 |