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 Cm-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.