This paper deals with a \(1/\kappa \)-type nonlocal flow for an initial convex closed curve \(\gamma _{0}\subset {\mathbb {R}}^{2}\) which preserves the convexity and the integral\(\ \int _{X\left( \cdot ,t\right) }\kappa ^{\alpha +1}ds,\ \alpha \in \left( -\infty ,\infty \right) ,\) of the evolving curve \(X\left( \cdot ,t\right) \). For\(\ \alpha \in [1,\infty ),\ \)it is proved that this flow exists for all time \(t\in [0,\infty )\) and \(X(\cdot ,t)\) converges to a round circle in \(C^{\infty }\) norm as \(t\rightarrow \infty \). For \(\alpha \in \left( -\infty ,1\right) \), a discussion on the possible asymptotic behavior of the flow is also given.

中文翻译：

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具有固定曲率积分的曲率半径驱动的非局部流

这种具有纸优惠\（1 / \卡帕\）型非局部流用于初始凸闭合曲线\（\伽马_ {0} \子集{\ mathbb {R}} ^ {2} \） ，其保留了凸和积分\（\ int_ {X \ left（\ cdot，t \ right）} \ kappa ^ {\ alpha +1} ds，\ \ alpha \ in \ left（-\ infty，\ infty \ right） ，\）的变化曲线\（X \ left（\ cdot，t \ right）\）。对于\（\ \ alpha \ in [1，\ infty），\ \），证明该流在所有时间\（t \ in [0，\ infty）\）和\（X（\ cdot，t ）\）以\（t \ rightarrow \ infty \）收敛到\（C ^ {\ infty} \）范数中的圆。对于\（\ alpha \ in \ left（-\ infty，1 \ right）\），还讨论了流动的可能渐近行为。