# Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $$\chi(t,s)$$ is a curve in $$\mathbb R^3$$, be derived from the Euler equation $$\partial_t \omega + v\cdot \nabla \omega = \omega \cdot \nabla v, \quad \operatorname{div} v = 0,$$ where $$v:\mathbb R \times \mathbb R^3 \to \mathbb R^3$$, and $$\omega = \operatorname{curl}(v):\mathbb R \times \mathbb R^3 \to \mathbb R^3.$$

I've asked a more general question at Survey on the vortex filament equation.

• you can find a detailed derivation in arxiv.org/abs/1603.00227 . May 15 '19 at 20:11
• @CarloBeenakker Thank you. That's very helpful.
– Kei
May 15 '19 at 21:02