Open Access
2015 Existence of knotted vortex tubes in steady Euler flows
Alberto Enciso, Daniel Peralta-Salas
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Acta Math. 214(1): 61-134 (2015). DOI: 10.1007/s11511-015-0123-z


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 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.


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Alberto Enciso. Daniel Peralta-Salas. "Existence of knotted vortex tubes in steady Euler flows." Acta Math. 214 (1) 61 - 134, 2015.


Received: 5 November 2012; Revised: 6 October 2014; Published: 2015
First available in Project Euclid: 30 January 2017

zbMATH: 1317.35184
MathSciNet: MR3316756
Digital Object Identifier: 10.1007/s11511-015-0123-z

Primary: 35Q31
Secondary: 35J25 , 37C55 , 37J40 , 37N10 , 57M25

Keywords: Beltrami fields , Euler equation , invariant tori , KAM theory , knots , Runge-type approximation

Rights: 2015 © Institut Mittag-Leffler

Vol.214 • No. 1 • 2015
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