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

Abstract

AbstractWe consider the Euler equations in $$\mathbb R^3$$ R 3 expressed in vorticity form $$\begin{aligned} \left\{ \begin{array}{l} \vec \omega _t + (\mathbf{u}\cdot {\nabla } ){\vec \omega } =( \vec \omega \cdot {\nabla } ) \mathbf{u} \\ \mathbf{u} = \mathrm{curl}\vec \psi ,\ -\Delta \vec \psi = \vec \omega . \end{array}\right. \end{aligned}$$ ω → t + ( u · ∇ ) ω → = ( ω → · ∇ ) u u = curl ψ → , - Δ ψ → = 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.