Dynamics of Elastic Wires: Preserving Area without Nonlocality

TL;DR

This paper introduces a novel sixth-order elastic curve evolution that preserves enclosed area, proves its global existence, and demonstrates convergence to critical points when penalizing length.

Contribution

It derives a new $H^{-1}$-gradient flow for elastic energy that maintains area and establishes its global existence and convergence properties.

Findings

Proves global existence of the area-preserving elastic flow.

Shows convergence to critical points with length penalization.

Introduces a sixth-order evolution equation for planar curves.

Abstract

We derive an $H^{-1}$-gradient flow of the elastic energy which preserves the enclosed area of evolving planar curves. For this new sixth-order evolution equation, we prove a global existence result. Additionally, by penalizing the length, we show convergence to an area constrained critical point of the elastic energy.