Regularization for point vortices on $\mathbb S^2$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913579977932800 |
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| author | Sakajo, Takashi Zou, Changjun |
| author_facet | Sakajo, Takashi Zou, Changjun |
| contents | We construct a series of patch type solutions for incompressible Euler equation on $\mathbb S^2$, which constitutes the regularization for steady or traveling point vortex systems. We first prove the existence of $k$-fold symmetric patch solutions, whose limit is the well-known von Kármán point vortex street on $\mathbb S^2$; then we consider the general steady case, where besides a non-localized part induced by the sphere rotation, $j$ positive and $k$ negative patches are located near a nondegenerate critical point of the Kirchhoff--Routh function on $\mathbb S^2$. Our construction is accomplished by Lyapunov--Schmidt reduction argument, where the traveling speed or vortex patch location are used to eliminate the degenerate direction of a linearized operator. We also show that the boundary of each vortex patch is a $C^1$ close curve, which is a perturbation of a small ellipse in the spherical coordinates. As far as we know, this is the first attempt for a regularization of the point-vortex equilibria on $\mathbb S^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11388 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularization for point vortices on $\mathbb S^2$ Sakajo, Takashi Zou, Changjun Analysis of PDEs 76B47 We construct a series of patch type solutions for incompressible Euler equation on $\mathbb S^2$, which constitutes the regularization for steady or traveling point vortex systems. We first prove the existence of $k$-fold symmetric patch solutions, whose limit is the well-known von Kármán point vortex street on $\mathbb S^2$; then we consider the general steady case, where besides a non-localized part induced by the sphere rotation, $j$ positive and $k$ negative patches are located near a nondegenerate critical point of the Kirchhoff--Routh function on $\mathbb S^2$. Our construction is accomplished by Lyapunov--Schmidt reduction argument, where the traveling speed or vortex patch location are used to eliminate the degenerate direction of a linearized operator. We also show that the boundary of each vortex patch is a $C^1$ close curve, which is a perturbation of a small ellipse in the spherical coordinates. As far as we know, this is the first attempt for a regularization of the point-vortex equilibria on $\mathbb S^2$. |
| title | Regularization for point vortices on $\mathbb S^2$ |
| topic | Analysis of PDEs 76B47 |
| url | https://arxiv.org/abs/2411.11388 |