Properties And Examples Of Curve Flows In Plane

Studies on curve flows have been very important and active.Let X(u,t)=(x(u,t),y(u,t)):[a,b]×[0,?)?R2 be a family of planar closed curves.The classical curve shortening flow is:(?)X/(?)t= kN,where,k = k(u,t)is the curvature of the evolving curve,N = N(u,t)is the inward normal vector.Hamilton-Gage has proved that convex curve will shrink to a round point in finite time.Recently many scholars studied curve flows of other types,for instance area preserving or length preserving curve flows.Inspired by the work of Pan Shengliang,we studied two examples of curve flows in this paper.First,we consider the following flow:Xt(k2-??/A-(1-?)L2-2A?/2A2)N,0???1X(u,0)= X0(u)We prove that the curve remains convex during the evolution and converges to a finite circle smoothly.Then we consider the following flow:(?)X(u,t)/(?)t=(p(u,t)-?L/2?-(1-?)1/k)N(u,t)X(u,0)(?)= X0(u)where 0 ???1.Then for all t ?[0,+?),the curve flow has global solution.The curves remain convex with length preserved while the enclosed area increasing.The curve becomes more and more circular and converges to a circle in C0 norm.