This paper deals with the $1/{\kappa }^{\alpha }$-type area-preserving nonlocal flow of smooth convex closed plane curves for all constant $\alpha >0$. Under this flow, the convexity of the evolving curve is preserved. Due to the existence of finite time curvature blow-up examples, it is shown that, if the curvature κ will not blow up in finite time, the evolving curve will converge smoothly to a circle as $t\to \infty$.

中文翻译：

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在保面积上凸封闭平面曲线的反曲率流

本文涉及 $\mathrm{1个}/{\kappa }^{\alpha }$所有常数的光滑凸封闭平面曲线的保型非局部流 $\alpha >0$。在这种流动下，演化曲线的凸度得以保留。由于存在有限的时间曲率爆破实例，因此表明，如果曲率κ在有限的时间内不爆炸，则演化曲线将平滑地收敛为一个圆。$Ť\to \infty$。