Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows

Abstract

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.

Global dynamics of a parabolic type equation arising from the curvature flow

“…Note that γ 0 is axis-symmetric and has 2mπ total curvature. Then the problem (1.1) could be used to describe the motion of γ 0 driven by a lengthpreserving curvature flow, see related studies in [15,17,19] and references therein. In particular, when a = π, γ 0 is a simple closed convex curve and the flow converges to a round circle as time goes to infinity according to the result in [17].…”