On the motion of a curve by its binormal curvature

TL;DR

This paper introduces a weak formulation for the binormal curvature flow of curves in three-dimensional space, allowing for initial data as integral currents and establishing global existence and uniqueness under certain conditions.

Contribution

It develops a broad weak formulation for the binormal curvature flow that includes integral currents as initial data and proves global existence and weak-strong uniqueness.

Findings

Weak formulation accommodates integral currents as initial data.

Global existence of solutions is established.

Weak-strong uniqueness holds when self-intersections are absent.

Abstract

We propose a weak formulation for the binormal curvature flow of curves in $\R^3.$ 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.