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Existence of knotted vortex tubes in steady Euler flows

Alberto Enciso Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas Daniel Peralta-Salas Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas

Dynamical Systems mathscidoc:1701.11007

Acta Mathematica, 214, (1), 61-134, 2012.11
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.
Euler equation; invariant tori; KAM theory; knots; Beltrami fields; Runge-type approximation
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@inproceedings{alberto2012existence,
  title={Existence of knotted vortex tubes in steady Euler flows},
  author={Alberto Enciso, and Daniel Peralta-Salas},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203404173288799},
  booktitle={Acta Mathematica},
  volume={214},
  number={1},
  pages={61-134},
  year={2012},
}
Alberto Enciso, and Daniel Peralta-Salas. Existence of knotted vortex tubes in steady Euler flows. 2012. Vol. 214. In Acta Mathematica. pp.61-134. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203404173288799.
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