Stability of vortex quadrupoles with odd-odd symmetry
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| author | Choi, Kyudong Jeong, In-Jee Yao, Yao |
| author_facet | Choi, Kyudong Jeong, In-Jee Yao, Yao |
| contents | For the 2D incompressible Euler equations, we establish global-in-time ($t \in \mathbb{R}$) stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is signed, sufficiently concentrated and close to its radial rearrangement up to a translation in $L^1$, we prove that it remains so for all times.
The main difficulty is that the kinetic energy maximization problem in a quadrant -- the typical approach for establishing vortex stability -- lacks a solution, as the kinetic energy continues to increase when the vorticity escapes to infinity. We overcome this by taking dynamical information into account: finite-time desingularization result is combined with monotonicity of the first moment and a careful analysis of the interaction energies between vortices. The latter is achieved by new pointwise estimates on the Biot--Savart kernel and quantitative stability results for general interaction kernels. Moreover, with a similar strategy we obtain stability of a pair of opposite-signed Lamb dipoles moving away from each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19822 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stability of vortex quadrupoles with odd-odd symmetry Choi, Kyudong Jeong, In-Jee Yao, Yao Analysis of PDEs Mathematical Physics For the 2D incompressible Euler equations, we establish global-in-time ($t \in \mathbb{R}$) stability of vortex quadrupoles satisfying odd symmetry with respect to both axes. Specifically, if the vorticity restricted to a quadrant is signed, sufficiently concentrated and close to its radial rearrangement up to a translation in $L^1$, we prove that it remains so for all times. The main difficulty is that the kinetic energy maximization problem in a quadrant -- the typical approach for establishing vortex stability -- lacks a solution, as the kinetic energy continues to increase when the vorticity escapes to infinity. We overcome this by taking dynamical information into account: finite-time desingularization result is combined with monotonicity of the first moment and a careful analysis of the interaction energies between vortices. The latter is achieved by new pointwise estimates on the Biot--Savart kernel and quantitative stability results for general interaction kernels. Moreover, with a similar strategy we obtain stability of a pair of opposite-signed Lamb dipoles moving away from each other. |
| title | Stability of vortex quadrupoles with odd-odd symmetry |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2409.19822 |