Dynamics of embedded curves by doubly-nonlocal reaction–diffusion systems

Abstract

We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of complexity for embedded curves. This line of investigation culminates in a family of complexity bounds that relate a rather broad class of models to a generalized, or weighted, variant of the crossing number. Our dynamic results include global well-posedness of the associated partial differential equations, regularity of equilibria for these flows as well as a more detailed investigation of dynamics near such equilibria. Finally, we explore a few global dynamical properties of these models numerically.

Sobolev Gradients for the Möbius Energy

“…Both theoretical and numerical results have been obtained on linear combinations of the bending energy and the Möbius energy [51,61,95]. More generally, in order to find minimizers of an elastic energy within an isotopy class, each knot energy can be employed in two ways: either as regularizer as it was done, e.g., in [4][5][6]29,32,40,94,97], or by using it to encode a hard bound into the domain, which was done with the knot thickness in [39,76,96]. Fig.…”

A speed preserving Hilbert gradient flow for generalized integral Menger curvature

“…The most significant contributions are the long time existence results of Blatt for the L 2 -flow of O'Hara's knot energies including the Möbius energy [7,10,9]. There is also work on the L 2flows of linear combinations of the classic bending energy and a (non-local) self-avoidance term [24], [36], but there the crucial a priori estimates are obtained by virtue of the leading order curvature energy. For a linear combination of a discrete bending and self-repulsive tangentpoint-type energy, however, numerical analysis has been performed recently by S. Bartels, Reiter, and J. Riege [5,4].…”