Existence of knotted vortex tubes in steady Euler flows

TL;DR

This paper proves that complex knotted and linked vortex tubes can exist as steady solutions in the incompressible Euler equations, with rich vortex line structures including invariant tori and periodic lines.

Contribution

It demonstrates the existence of steady Euler flows with knotted vortex tubes, extending classical ideas and constructing explicit vortex configurations.

Findings

Existence of knotted and linked vortex tubes in steady Euler flows.

Vortex lines form a rich structure with invariant tori and periodic lines.

Constructs vortex tubes that tend to zero at infinity.

Abstract

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R^3. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R^3, we show that they can be transformed with a C^m-small diffeomorphism into a set of vortex tubes of a Beltrami field that tends to zero at infinity. The structure of the vortex lines in the tubes is extremely rich, presenting a positive-measure set of invariant tori and infinitely many periodic vortex lines. The problem of the existence of steady knotted vortex tubes can be traced back to Lord Kelvin.