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arXiv · 2007.00606

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

Juan Dávila·Manuel del Pino·Monica Musso·Juncheng Wei
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Abstract

We consider the Euler equations in ${\mathbb R}^3$ expressed in vorticity form. 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 {\em 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 similar helical filaments travelling and rotating together.

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Juan Dávila, Manuel del Pino, Monica Musso, Juncheng Wei. 2020-07-01. Travelling helices and the vortex filament conjecture in the incompressible Euler equations. https://arxiv.org/abs/2007.00606

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