Existence of knotted vortex tubes in steady Euler flows

Abstract

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in ${\mathbb{R}}^3$. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in ${\mathbb{R}}^3$, 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 thin vortex tubes can be traced back to Lord Kelvin.