On some simple examples of non-parabolic curve flows in the plane

We discuss several examples of non-parabolic curve flows in the plane. In these flows, the speed functions do not involve the curvature at all. Although elementary in nature, there are some interesting properties. In particular, certain non-parabolic flows can be employed to evolve a convex closed curve to become circular or to evolve a non-convex curve to become convex eventually, like what we have seen in the classical curve shortening flow (parabolic flow) by Gage and Hamilton (J Differ Geom 23:69–96, 1986), Grayson (J Differ Geom 26:285–314, 1987).