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We provide sufficient conditions on an initial curve for the area preservingand the length preserving curvature flows of curves in a plane, to develop asingularity at some finite time or converge to an $m$-fold circle as time goesto infinity. For the area-preserving flow, the positivity of the enclosedalgebraic area determines whether the curvature blows up in finite time or not,while for the length-preserving flow, it is the positivity of an energyassociated with initial curve that plays such a role.

Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows