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

Abstract

We consider the Euler equations in mathbb R^3 expressed in vorticity form ω→t+(u·∇)ω→=(ω→·∇)uu=curlψ→,-Δψ→=ω→.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\vec \\omega _t + (\\mathbf{u}\\cdot {\ abla } ){\\vec \\omega } =( \\vec \\omega \\cdot {\ abla } ) \\mathbf{u} \\\\ \\mathbf{u} = \\mathrm{curl}\\vec \\psi ,\\ -\\Delta \\vec \\psi = \\vec \\omega . \\end{array}\\right. \\end{aligned}$$\\end{document}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.