The gradient flow for entropy on closed planar curves

TL;DR

This paper studies the gradient flow of entropy on closed planar curves, showing that it causes curves to expand and converge smoothly to a round circle, regardless of initial shape.

Contribution

It proves convergence of the entropy gradient flow to a round circle for various initial convex and embedded curves, extending understanding of curve evolution.

Findings

Curves expand with radius growing like the square root of time.

Flow converges smoothly to a round, multiply-covered circle.

Results apply to both convex and certain non-convex initial curves.

Abstract

In this paper we consider the steepest descent L2-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class $C^2$ or embedded of class $W^{2,2}$ bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.