Mean field limit of interacting filaments for 3D Euler equations

Solutions to the 3D Euler equations are obtained as a mean field limit of finite families of interacting curves, the so called vortex filaments. Families of N interacting curves are considered, with long range, mean field type interaction. A family of curves defines a 1-current, concentrated on the curves, analog of the empirical measure of interacting point particles. This current is proved to converge, as N goes to infinity, to a solution of the 3D Euler equation. In the limit, each curve interacts with the mean field current and two different curves have an independence property if they are independent at time zero.