Abstract: Two kinds of convex curve flows on a plane were studies. One is combination of an area-preserving curve flow proposed and a length-preserving curve flow proposed, this flow reduces the curve length but increases the enclosed area in the evolution process, the other is convex combination of the length-preserving curve flows, it keeps the length constant and expands the area. The two curvature flows exist globally and converge to a circle in the C^∞ metric as time goes to infinity.

Key words: C^∞ metric, convex curve, curvature flow, support function