# Curve-shortening flow of open, elastic curves in $\mathbb{R}^2$ with repelling endpoints: A minimizing movement approach

## TL;DR

This paper investigates the evolution of open elastic curves in the plane with repelling endpoints using a gradient flow approach, establishing long-term existence and supporting findings with numerical experiments.

## Contribution

It introduces a novel gradient flow model for elastic curves with Coulomb repulsion and proves its long-time existence via a minimizing movement scheme.

## Key findings

- Long-time existence of the flow is proven.
- Numerical experiments support theoretical results.
- The model captures the behavior of elastic curves with endpoint repulsion.

## Abstract

We study an $L^{2}$-type gradient flow of an immersed elastic curve in $\mathbb{R}^{2}$ whose endpoints repel each other via a Coulomb potential. By De Giorgi's minimizing movements scheme we prove long-time existence of the flow. The work is complemented by several numerical experiments.