Research On Two Types Of Curve Flow Problems In The Plane

This paper mainly investigates two kinds of convex curve flows in the plane.Firstly,we consider the following evolving problem:Let X(u,t):[a,b]×[0,T)?R2 be a family of closed plane curves.(?)X(u,t)/(?)=(k2-??/A-(1-?)4?2/L2)N,1/2???1(3)X(u,0)=X0(u)We will prove the solutions to this curve flow exist for all time,the curves remain convex,and the length of the curves and the enclosed area are decreasing with uniform positive lower bounds.Furthermore the evolving curves converge to round circles in C? sense as t??.Next we discuss one kind of curve expanding flow.Suppose F=F(u,t):[a,b]×[0,T)?R2 is a family of closed plane curves.(?)F(u,t)/(?)t=(ap+(1-a)?(t)-1/k)N(u,t)(a?0)(4)Then we will prove that for a certain range of functions ?(t),the cir-cumference L will increase or decrease correspondingly,but the area enclosed by the curve will always increase.With the help of the sup-port function,we will obtain its global existence.Finally,the evolving curves converge to round circles in C0 sense as t??.