  In this paper we consider the steepest descent $L^2$-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 convex of class $C^2$ or
embedded of class $W^{2,2}$ bounding a convex domain), the flow converges
smoothly to a round expanding multiply-covered circle.
