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

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.

32 pages