On the motion of a curve by its binormal curvature

We propose a weak formulation for the binormal curvature flow of curves in ${\mathbb{R}}^3\mathrm{.}$ This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.