On Area-preserving and Length-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\geq1.\$Comment: Published in 'Calculus of Variations and Partial Differential Equations