Convergence to equilibrium of gradient flows defined on planar curves

TL;DR

This paper studies the long-term behavior of gradient flows on planar curves, proving convergence to stationary solutions under certain conditions and exploring applications of these results.

Contribution

It establishes convergence of gradient flows on planar curves to stationary solutions when stationary sets are finite, with practical applications.

Findings

Flow solutions converge to stationary solutions over time.

Finite stationary solution sets guarantee convergence.

Applications demonstrate the theoretical results' relevance.

Abstract

We consider the evolution of open planar curves by the steepest descent flow of a geometric functional, under different boundary conditions. We prove that, if any set of stationary solutions with fixed energy is finite, then a solution of the flow converges to a stationary solution as time goes to infinity. We also present a few applications of this result.