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Abstract
We consider the Euler equations in R ^{3} expressed in vorticity form {ω→t+(u·∇)ω→=(ω→·∇)uu=curlψ→,Δψ→=ω→.A classical question that goes back to Helmholtz is to describe the evolution of solutions with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the socalled binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a longstanding open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortexrings. We provide what appears to be the first rigorous construction of helical filaments, associated to a translatingrotating helix. The solution is defined at all times and does not change form with time. The result generalizes to multiple polygonal helical filaments travelling and rotating together.
Original language  English 

Article number  119 
Journal  Calculus of Variations and Partial Differential Equations 
Volume  61 
Issue number  4 
Early online date  5 May 2022 
DOIs  
Publication status  Published  31 Aug 2022 
ASJC Scopus subject areas
 Analysis
 Applied Mathematics
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Dive into the research topics of 'Travelling helices and the vortex filament conjecture in the incompressible Euler equations'. Together they form a unique fingerprint.Projects
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Concentration phenomena in nonlinear analysis
Engineering and Physical Sciences Research Council
27/04/20 → 26/04/23
Project: Research council