The gradient flow of the potential energy on the space of arcs

Abstract

We study the gradient flow of the potential energy on the infinite-dimensional Riemannian manifold of spatial curves parametrized by the arc length, which models overdamped motion of a falling inextensible string. We prove existence of generalized solutions to the corresponding nonlinear evolutionary PDE and their exponential decay to the equilibrium. We also observe that the system admits solutions backwards in time, which leads to non-uniqueness of trajectories.

Uniformly Compressing Mean Curvature Flow

“…We stress that the theorem does not cover Theorem 2 due to the relaxation of the unit speed constraint in (68). The proof mimicks the one of [46,Theorem 3] and has the following outline. We rewrite (68) as a first-order system, and approximate it by Hilbertian gradient flows.…”

“…with respect to the L 2 (S 1 ; R d )-induced metric (see also Appendix B and our recent work [46] which analyzes the gradient flow of the potential energy on a space very similar toÃ).…”

“…The normalized flow (19)- (20) can be interpreted in the spirit of [46] as an overdamped motion of an inextensible loop whose particles are repelled from the origin with the force equal to the radius vector.…”

“…Surprisingly enough, our normalized flow is also a gradient flow: namely, the positive gradient flow of the L 2 -mass on the space of volume-preserving immersions. Our recent work [46] studies the gradient flow of a different functional (potential energy) on a similar Riemannian structure, which turns out to be a model for an overdamped fall of an inextensible string in a gravitational field. A similar mechanical interpretation for our normalized UCMCF is an overdamped motion of an inextensible loop (k = 1) or an incompressible membrane (k > 1) repelled from the origin with the force field identically equal to the radius vector.…”

“…Indeed, since c(t), t ∈ [0, t 0 ], is bounded away from zero and has uniformly bounded modulus (cf. Corollary 3.14), the solution to(46) can be extended for longer time as long as ξ(t, ·) C 2+α (S 1 ) remains bounded.…”