A family of evolution equations connecting area-preserving to length-preserving curve flows

In this paper we define a one parameter family of curve flows in the plane connecting a type of area-preserving to length-preserving curve flows. When the initial curve is closed and convex, we show that along the flows the length of the curve is non-increasing while the enclosed area is non-decreasing. We show that the solutions exist for all time and converge to a circle in C0 norm when t→+∞.