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

TL;DR

The paper constructs clustered, vortex-like solutions to a 2D divergence-form semilinear elliptic equation with a logarithmic forcing, using a sharp $C^1$ expansion of Green's functions and a finite-dimensional reduction. The main result is the existence of solutions with a cluster point at the origin and cluster distance $|\ln\varepsilon|^{-1/2}$; the small-scale structure is governed by a reduced energy $H_N$ determined by the coefficient matrix $K$ and the function $q$. These clustered states are then applied to justify and generalize traveling-rotating helical vortex filaments in 3D Euler flows within infinite cylinders, yielding slender co-rotating helices and a variety of asymmetric configurations with arbitrary circulations. The approach combines linear and nonlinear solvability theory, precise energy expansions, and symmetry reductions to reduce the infinite-dimensional Euler dynamics to finite-dimensional critical-point problems. The results provide a rigorous bridge between the vortex-filament model and the PDE framework, extending prior work and enabling new families of nearly parallel helical filaments to be realized mathematically.

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 $ \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.

Theorems & Definitions (44)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10