The Gradient Flow of O'Hara's Knot Energies

TL;DR

This paper investigates the gradient flow of O'Hara's knot energies, establishing short and long-term existence and convergence to critical points for a specific family of these energies.

Contribution

It provides the first rigorous analysis of the gradient flow of O'Hara's knot energies, including existence, convergence, and characterization of critical points.

Findings

Short time existence of the flow

Long time existence and convergence

Flow converges to critical points

Abstract

Jun O'Hara invented a family of knot energies $E^{j,p}$, $j,p \in (0, \infty)$. We study the negative gradient flow of the sum of one of the energies $E^\alpha = E^{\alpha,1}$, $\alpha \in (2,3)$, and a positive multiple of the length. Showing that the gradients of these knot energies can be written as the normal part of a quasilinear operator, we derive short time existence results for these flows. We then prove long time existence and convergence to critical points.