Co-rotating nearly parallel helical vortices with small cross-section in 3D incompressible Euler equations

Abstract

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 $\epsilon$. 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 $|\ln \epsilon {|}^{-\frac{1}{2}}$ 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 $\epsilon$ and arbitrary circulation $\hat{I}º$ and concentrates near ``$2N$'' and ``$2N+1$'' type of co-rotating helical solutions of nearly parallel vortex filaments model as $\epsilon \to 0$, 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'', ``$2\times 2$ asymmetric'' and ``$2\times 2+1$ asymmetric'' type of co-rotating helical filaments are also considered.

55 pages