Three Evolution Problem For Plane Curves

Through careful study of previous articles,this paper studies three kinds of curve flows.First we discuss a curve expanding flow:(?)We prove that when the initial curve is a closed convex curve,the curve flow remains convex during the development process,and the perimeter of the curve increases with time.The enclosed area is increasing,and it is proved that under the C~0 norm,when time t?+?,the limit curve converges to a circle with a finite radius.Secondly,we discuss another curve expanding flow:(?)We prove that the perimeter of the curve and the area enclosed by the curve are monotonically increasing with time,and the isoperimetric inequality of the curve flow is monotonously decreasing with time.Under the C~0 norm,when t?+?,the limit curve converges to a finite circle.Finally,a curve combined flow is discussed:(?)The curve flow is composed with area-preserving curve flow and length-preserving curve flow.We prove that during the development of the curve flow,the perimeter of the curve decreases with time,and the area enclosed by the curve increases with time.When t?+?,under the C~0 norm,the limit curve converges to a finite circle.