Existence of knotted vortex tubes in steady Euler flows

2012/10/23 by Alberto Enciso, Daniel Peralta‐Salas, Enciso, Alberto +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1210.6271

openalex publication_date 2012/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible Euler equation in R³. More precisely, given a finite collection of (possibly linked and knotted) disjoint thin tubes in R³, 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.

Cited by

• On approximation theorems for solutions to strongly parabolic systems in anisotropic Sobolev spaces
• Exact Lagrangian Realization and Robust Strain Sensing in Incompressible Flow

Related

• Contact topology and hydrodynamics III: knotted flowlines
• Knotted structures in high-energy Beltrami fields on the torus and the\n sphere
• Helical vortex filaments with compactly supported cross-sectional vorticity for the incompressible Euler equations in ℝ³
• Confined Vortex Surface and Irreversibility. 3. Nested Tubes and Energy Cascade
• Travelling helices and the vortex filament conjecture in the incompressible Euler equations