Crystalline elastic flow of polygonal curves: long time behaviour and convergence to stationary solutions

TL;DR

This paper studies the long-term behavior of crystalline elastic flow of polygonal curves with crystalline anisotropy, proving existence, convergence to stationary solutions, and classifying stationary and translating solutions.

Contribution

It introduces a framework for analyzing the crystalline elastic flow of polygonal curves, including existence, continuation, and convergence results, with classifications of stationary solutions.

Findings

Unique regular flow exists until segments with zero crystalline curvature disappear.

Flow can be restarted finitely many times, maintaining the curve's index.

Flow converges to stationary solutions as time approaches infinity.

Abstract

Given a planar crystalline anisotropy, we study the crystalline elastic flow of immersed polygonal curves, possibly also unbounded. Assuming that the segments evolve by parallel translation (as it happens in the standard crystalline curvature flow), we prove that a unique regular flow exists until a maximal time when some segments having zero crystalline curvature disappear. Furthermore, for closed polygonal curves, we analyze the behaviour at the maximal time, and show that it is possible to restart the flow finitely many times, yielding a globally in time evolution, that preserves the index of the curve. Next, we investigate the long-time properties of the flow using a Lojasiewicz-Simon-type inequality, and show that, as time tends to infinity, the flow fully converges to a stationary curve. We also provide a complete classification of the stationary solutions and a partial classification of the translating solutions in the case of the square anisotropy.