Co-rotating nearly parallel helical vortices with small cross-section in 3D incompressible Euler equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918192183508992 |
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| author | Cao, Daomin Wan, Jie |
| author_facet | Cao, Daomin Wan, Jie |
| contents | In this article, we consider clustered solutions to a semilinear elliptic equation in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in Ω,\\ u=0,\ \ &x\in\partial Ω\end{cases} \end{equation*} for small values of $ \varepsilon $. Using Green's function of the elliptic operator $ -\text{div}(K(x)\nabla) $ and finite-dimensional reduction method, we prove that there exist clustered solutions with cluster point $ 0 $ and cluster distance $ |\ln\varepsilon| ^{-\frac{1}{2}} $ whose small-structure is governed by some functional $ H_N $ determined by $ K $ and $ q $. As an application, we prove the existence of traveling-rotating helical vorticity fields to 3D incompressible Euler equations in infinite cylinders, whose support sets consist of helical tubes with small cross-section of radius $ \varepsilon $ and arbitrary circulation $ κ$ and concentrates near ``$ 2N $'' and ``$ 2N+1 $'' type of co-rotating helical solutions of nearly parallel vortex filaments model as $ \varepsilon\to0 $, which justifies the result in Klein, Majda and Damodaran [1995, JFM] and generalizes results in Guerra and Musso [arxiv: 2502.01470]. Several kinds of solutions such as ``2 asymmetric'', ``$ 2\times2 $ asymmetric'' and ``$ 2\times2+1 $ asymmetric'' type of co-rotating helical filaments are also considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Co-rotating nearly parallel helical vortices with small cross-section in 3D incompressible Euler equations Cao, Daomin Wan, Jie Analysis of PDEs In this article, we consider clustered solutions to a semilinear elliptic equation in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in Ω,\\ u=0,\ \ &x\in\partial Ω\end{cases} \end{equation*} for small values of $ \varepsilon $. Using Green's function of the elliptic operator $ -\text{div}(K(x)\nabla) $ and finite-dimensional reduction method, we prove that there exist clustered solutions with cluster point $ 0 $ and cluster distance $ |\ln\varepsilon| ^{-\frac{1}{2}} $ whose small-structure is governed by some functional $ H_N $ determined by $ K $ and $ q $. As an application, we prove the existence of traveling-rotating helical vorticity fields to 3D incompressible Euler equations in infinite cylinders, whose support sets consist of helical tubes with small cross-section of radius $ \varepsilon $ and arbitrary circulation $ κ$ and concentrates near ``$ 2N $'' and ``$ 2N+1 $'' type of co-rotating helical solutions of nearly parallel vortex filaments model as $ \varepsilon\to0 $, which justifies the result in Klein, Majda and Damodaran [1995, JFM] and generalizes results in Guerra and Musso [arxiv: 2502.01470]. Several kinds of solutions such as ``2 asymmetric'', ``$ 2\times2 $ asymmetric'' and ``$ 2\times2+1 $ asymmetric'' type of co-rotating helical filaments are also considered. |
| title | Co-rotating nearly parallel helical vortices with small cross-section in 3D incompressible Euler equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.05956 |