Travelling helices and the vortex filament conjecture in the incompressible Euler equations
Juan Dávila, Manuel del Pino, Monica Musso, Juncheng Wei
Abstract
Abstract We consider the Euler equations in $$\mathbb R^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:math> expressed in vorticity form $$\begin{aligned} \left\{ \begin{array}{l} \vec \omega _t + (\mathbf{u}\cdot {\nabla } ){\vec \omega } =( \vec \omega \cdot {\nabla } ) \mathbf{u} \\ \mathbf{u} = \mathrm{curl}\vec \psi ,\ -\Delta \vec \psi = \vec \omega . \end{array}\right. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mfenced><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:msub><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>·</mml:mo><mml:mi>∇</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>·</mml:mo><mml:mi>∇</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mrow/><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>curl</mml:mi><mml:mover><mml:mi>ψ</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:mspace/><mml:mo>-</mml:mo><mml:mi>Δ</mml:mi><mml:mover><mml:mi>ψ</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mover><mml:mi>ω</mml:mi><mml:mo>→</mml:mo></mml:mover><mml:mo>.</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mfenced></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math> 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.