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