On length-preserving and area-preserving nonlocal flow of convex closed plane curves

For any $$\alpha >0,$$ we study $$k^{\alpha }$$ -type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in $$C^{\infty } $$ -norm. Other relevant $$k^{\alpha }$$ -type nonlocal flow is also discussed when $$\alpha \ge 1.$$