Travelling helices and the vortex filament conjecture in the incompressible Euler equations
Abstract
We consider the Euler equations in 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.
Keywords
Cite
@article{arxiv.2007.00606,
title = {Travelling helices and the vortex filament conjecture in the incompressible Euler equations},
author = {Juan Dávila and Manuel del Pino and Monica Musso and Juncheng Wei},
journal= {arXiv preprint arXiv:2007.00606},
year = {2020}
}