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

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.