Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompressible Euler equations
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866909683744243712 |
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| author | Cao, Daomin Fan, Junhong Qin, Guolin Wan, Jie |
| author_facet | Cao, Daomin Fan, Junhong Qin, Guolin Wan, Jie |
| contents | In this paper, we investigate the time evolution of helical vortices without swirl for the incompressible Euler equations in $\mathbb R^3$ under general initial assumptions. Assume the initial helical vorticity is sharply concentrated in $N$ distinct $\ep$-neighborhoods, whose mutual distances vanish as $O(1/|\ln \ep|)$, and each vortex core possesses vorticity mass of order $1/|\ln \ep|^{1+b}$ for an arbitrary fixed $b\in\mathbb R$. We prove that as $\ep\to 0$, the motion of these helical vortices converges uniformly to a dynamical system derived herein over a time interval of order $1/|\ln\varepsilon|^{1-b}$. In the particular case $b=-1$, our results establish the evolution counterpart for interacting vortex helices constructed in [I. Guerra, M. Musso, Ann. Inst. H. Poincaré C Anal. Non Lináire, 2025]. Notably, for two interacting helical vortices with initial mutual distance $ ρ_0/|\ln \ep|$, by choosing $ρ_0$ sufficiently small, our analysis extends to timescales covering multiple periods. This result provides the first mathematical justification for the numerically observed phenomenon termed ``leapfrogging of Kelvin waves" reported in [N. Hietala et al., Phys. Rev. Fluids, 2016]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12240 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompressible Euler equations Cao, Daomin Fan, Junhong Qin, Guolin Wan, Jie Analysis of PDEs Primary: 76B47, Secondary: 37N10 In this paper, we investigate the time evolution of helical vortices without swirl for the incompressible Euler equations in $\mathbb R^3$ under general initial assumptions. Assume the initial helical vorticity is sharply concentrated in $N$ distinct $\ep$-neighborhoods, whose mutual distances vanish as $O(1/|\ln \ep|)$, and each vortex core possesses vorticity mass of order $1/|\ln \ep|^{1+b}$ for an arbitrary fixed $b\in\mathbb R$. We prove that as $\ep\to 0$, the motion of these helical vortices converges uniformly to a dynamical system derived herein over a time interval of order $1/|\ln\varepsilon|^{1-b}$. In the particular case $b=-1$, our results establish the evolution counterpart for interacting vortex helices constructed in [I. Guerra, M. Musso, Ann. Inst. H. Poincaré C Anal. Non Lináire, 2025]. Notably, for two interacting helical vortices with initial mutual distance $ ρ_0/|\ln \ep|$, by choosing $ρ_0$ sufficiently small, our analysis extends to timescales covering multiple periods. This result provides the first mathematical justification for the numerically observed phenomenon termed ``leapfrogging of Kelvin waves" reported in [N. Hietala et al., Phys. Rev. Fluids, 2016]. |
| title | Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompressible Euler equations |
| topic | Analysis of PDEs Primary: 76B47, Secondary: 37N10 |
| url | https://arxiv.org/abs/2505.12240 |