The Gradient Flow for Entropy on Closed Planar Curves

In this paper we consider the steepest descent L² L 2 -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 C² C 2 or embedded of class W2,2 W 2 , 2 bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.