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

TL;DR

This paper investigates the gradient flow of potential energy on the space of arc-length parametrized curves, modeling inextensible string dynamics, proving solution existence, decay to equilibrium, and exploring non-uniqueness in solutions.

Contribution

It introduces a mathematical framework for the gradient flow of potential energy on the space of curves, including existence and decay results, and highlights non-uniqueness of solutions.

Findings

Existence of generalized solutions to the nonlinear PDE.

Solutions decay exponentially to equilibrium.

Backward solutions lead to non-uniqueness.

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.