Clustered vortex helices with compactly supported cross-sectional vorticity in the 3D Euler equations

TL;DR

This paper constructs the first smooth multi-vortex solutions in 3D Euler equations with collapsing helical filaments and compactly supported cross-sectional vorticity, extending previous configurations.

Contribution

It introduces a novel gluing technique to create multi-vortex solutions with collapsing filaments and compact support in the entire space.

Findings

First smooth multi-vortex solution with collapsing filaments in 3D Euler equations.

Vorticity remains compactly supported in cross-section for all times.

Generalizes previous configurations with rapidly decaying vorticity cores.

Abstract

We consider the three-dimensional incompressible Euler equations for helical flows without swirl. By adapting gluing techniques, we construct the first smooth multi-vortex solution in the whole space $\mathbb{R}^3$ exhibiting a cluster of collapsing helical filaments, with the associated cross-sectional vorticity remaining compactly supported in $\mathbb{R}^2$ for all times. Our result generalises previous collapsing configurations in $\mathbb{R}^3$ with rapidly decaying vorticity cores, and extends related variational solutions obtained in infinite cylindrical domains.