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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.01399 |
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| _version_ | 1866916159165562880 |
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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*} in the whole space $\mathbb{R}^3$. Under the assumption that the initial velocity is helical and without swirl, we prove the global well-posedness of weak solutions in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^3)$. The vortex transport formula is also obtained in our article. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01399 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in $\mathbb{R}^3$ 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*} in the whole space $\mathbb{R}^3$. Under the assumption that the initial velocity is helical and without swirl, we prove the global well-posedness of weak solutions in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^3)$. The vortex transport formula is also obtained in our article. |
| title | Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in $\mathbb{R}^3$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2301.01399 |