The Gradient Flow of O'Hara's Knot Energies

Simon Blatt

The Gradient Flow of O'Hara's Knot Energies

Analysis of PDEs 2016-01-13 v1

Authors: Simon Blatt

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.

Keywords

knot energy gradient flow O'Hara energy quasilinear operator short-time existence long-time existence convergence

Cite

@article{arxiv.1601.02840,
  title  = {The Gradient Flow of O'Hara's Knot Energies},
  author = {Simon Blatt},
  journal= {arXiv preprint arXiv:1601.02840},
  year   = {2016}
}