No. 2022.06
Banach gradient flows for various families of knot energies
H. Matt, D. Steenebr├╝gge, and H. von der Mosel
Subject: Classical analysis and ODEs, differential geometry, optimization and control

Abstract

We establish long-time existence of Banach gradient flows for generalised integral Menger curvatures and tangent-point energies, and for O’Hara’s self-repulsive potentials $ E^{\alpha,p}$. In order to do so, we employ the theory of curves of maximal slope in slightly smaller spaces compactly embedding into the respective energy spaces associated to these functionals, and add a term involving the logarithmic strain, which controls the parametrisations of the flowing (knotted) loops. As a prerequisite, we prove in addition that O’Hara’s knot energies $E^{\alpha,p}$ are continuously differentiable.

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arXiv:2204.13603 accepted for publication in Journal of Evolution Equations