In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in the C∞C∞-topology to a simple circle. Our results yield a characterisation of the total amount of time during which the flow is not strictly convex, quantifying in a sense the failure of the maximum principle.
Parkins, S. & Wheeler, G. (2016). The polyharmonic heat flow of closed plane curves. Journal of Mathematical Analysis and Applications, 439 (2), 608-633.