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.

中文翻译：

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面积保持曲率流和长度保持曲率流中局部凸封闭曲线的演化

我们在初始曲线上为平面中的曲线的面积保留和长度保留曲率流提供了充分的条件，以在某个有限时间发展出奇异性，或者随着时间趋于无穷大而收敛到$ m $倍的圆。对于面积保持流，封闭的代数区的正性决定曲率是否在有限时间内爆炸，而对于长度保持流，与初始曲线相关的能量的正性起着这样的作用。