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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.09071 |
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| _version_ | 1866910367101222912 |
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| author | Guo, Dengjun Zhao, Lifeng |
| author_facet | Guo, Dengjun Zhao, Lifeng |
| contents | We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t Ω+U \cdot \nabla Ω-Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_0(x) \end{aligned}\right. \end{equation*} under the assumption that $Ω^z$ is helical and in the absence of vorticity stretching. Assuming that the initial vorticity $Ω_0$ is primarily concentrated within an $ε$ neighborhood of a helix $Γ_0$, we prove that its solution $Ω(\cdot,t)$ remain concentrated near a helix $Γ(t)$ for any $t \in [0,T)$, where $Γ(t)$ can be interpreted as $Γ_0$ rotating around the $x_3$ axis with a speed $V=C\log \frac{1}ε+O(1)$. It should be emphasized that the dynamics for the helical vortex filament are exhibited on the time interval $[0,T)$, which is longer than $\left[0, \frac{T}{\log\frac{1}ε}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long time dynamics for helical vortex filament in Euler flows Guo, Dengjun Zhao, Lifeng Analysis of PDEs We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t Ω+U \cdot \nabla Ω-Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_0(x) \end{aligned}\right. \end{equation*} under the assumption that $Ω^z$ is helical and in the absence of vorticity stretching. Assuming that the initial vorticity $Ω_0$ is primarily concentrated within an $ε$ neighborhood of a helix $Γ_0$, we prove that its solution $Ω(\cdot,t)$ remain concentrated near a helix $Γ(t)$ for any $t \in [0,T)$, where $Γ(t)$ can be interpreted as $Γ_0$ rotating around the $x_3$ axis with a speed $V=C\log \frac{1}ε+O(1)$. It should be emphasized that the dynamics for the helical vortex filament are exhibited on the time interval $[0,T)$, which is longer than $\left[0, \frac{T}{\log\frac{1}ε}\right)$. |
| title | Long time dynamics for helical vortex filament in Euler flows |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2403.09071 |