The evolution of a class of curve flows

In this paper, we study a fourth order curve flow ∂X∂t=−Δ2X for a smooth closed curve X in R2, which means that a fourth order nonlinear parabolic partial differential equation arises naturally in the field of biharmonic maps. We first derive the evolution equations for different geometric quantities along this flow. Utilizing the evolution equations, we prove that for any smooth closed initial curve in R2, the flow has a smooth solution for all time by assuming some additional conditions.