The Evolution of Nonlocal Curvature Flow Arising in a Hele--Shaw Problem

Abstract

We consider long time behavior of a given smooth convex embedded closed curve $\gamma_{0}\subset\mathbb{R}^{2}$ evolving according to a nonlocal curvature flow, which arises in a Hele--Shaw problem and has a prescribed rate of change in its enclosed area $A(t)$, i.e., $dA/dt=-\beta$, where $\beta\in(-\infty,\infty)$. Specifically, when the enclosed area expands at any fixed rate, i.e., $\beta\in(-\infty,0)$, or decreases at a fixed rate $\beta\in(0,2\pi)$, one has the round circle as the unique asymptotic shape of the evolving curves, while for a sufficiently large rate of area decrease, one can have $n$-fold symmetric curves (which look like regular polygons with smooth corners) as extinction shapes (self-similar solutions).