On the vortex filament conjecture for Euler flows

We consider an inviscid, incompressible fluid whose vorticity is concentrated along a smooth curve, the "vortex filament". Assuming that the fluid motion is described by the Euler equation, one may ask if there is a geometric law that describes the motion of the vortex filament. This question was answered in 1906 by da Rios, who formally showed that to leading order and modulo a rescaling in time, the curve evolves by an equation known as the binormal curvature flow.

In joint work with Bob Jerrard, we rediscover da Rios' result by means of rigorous arguments and under quite weak regularity assumtions on the vorticity.