Travelling helices and the vortex filament conjecture in the incompressible Euler equations

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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 so-called binormal curvature flow. Existence of true solutions concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. We provide what appears to be the first rigorous construction of helical filaments, associated to a translating-rotating 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 languageEnglish
Article number119
JournalCalculus of Variations and Partial Differential Equations
Volume61
Issue number4
Early online date5 May 2022
DOIs
Publication statusPublished - 31 Aug 2022

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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