Helical vortex filaments with compactly supported cross-sectional vorticity for the incompressible Euler equations in $\mathbb{R}^3$

TL;DR

This paper constructs smooth, helical vortex filaments with compactly supported cross-sectional vorticity for the 3D Euler equations, advancing the understanding of vortex filament dynamics and configurations.

Contribution

It provides the first explicit construction of smooth helical vortex filaments with compact support in the entire space, extending to multi-filament solutions and improving asymptotic analysis.

Findings

Constructed smooth helical vortex filaments with compact support.

Extended the construction to multi-filament configurations.

Provided detailed asymptotics for vorticity cores.

Abstract

We revisit the vortex filament conjecture for three-dimensional inviscid and incompressible Euler flows with helical symmetry and no swirl. Using gluing arguments, we provide the first construction of a smooth helical vortex filament in the whole space $\mathbb{R}^3$ whose cross-sectional vorticity is compactly supported in $\mathbb{R}^2$ for all times. The construction extends to a multi-vortex solution comprising several helical filaments arranged along a regular polygon. Our approach yields fine asymptotics for the vorticity cores, thus improving related variational results for smooth solutions in bounded helical domains and infinite pipes, as well as non-smooth vortex patches in the whole space.