# How do vortex filaments move?

In fluid dynamics (Navier-Stokes, incompressible case), a vortex filament, as I understand it, is a curve of points $$x\in\mathbb R^3$$ such that the velocity field $$v\colon \mathbb R^3\times\mathbb R\to\mathbb R^3$$ is parallel to its curl, $$\nabla \times v$$, also known as the vorticity. In other words, for each time $$t$$ we have a curve $$\gamma_t\colon (a,b)\to\mathbb R^3$$ that satisfies $$v(\gamma_t(s),t)=\lambda_{s,t}\nabla\times v(\gamma_t(s),t),$$ for some $$\lambda_{s,t}\in\mathbb R$$.

What is the equation governing the movement $$d\gamma_t/dt$$ of the vortex filament $$\gamma_t$$?