A log-type non-local flow of convex curves (Q2063730)

The present paper studies a log-type non-local flow of closed convex plane curves. Assume \(X(\varphi, t) = (x(\varphi, t), y(\varphi, t)) : S^1 \times [0, \omega) \rightarrow \mathbb R^2\) is a family of smooth strictly convex plane curves and consider the flow \begin{align*} \frac{\partial X}{\partial t}(\varphi,t)&=\ln\left(\kappa(\varphi,t)\sqrt\frac{A(t)}{\pi}\right)N(\varphi,t),\\ X(\varphi,0)&=X_0(\varphi),\varphi\in S^1, \end{align*} where \(X_0(\varphi)\) is a simple closed strictly convex plane curve, \(A(t)\) the area enclosed by the evolving curve, \(\kappa >0\) the curvature and \(N(t)\) the inward pointing unit normal vector to the evolving curve. The main result is the following: Theorem 1: A closed convex plane curve evolving under the above flow keeps convex, shortens its length while expands the enclosed area and converges to a finite circle as t approaches infinity in the \(C^{\infty}\) metric.