  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.
