Invariant Geometric Properties in Hele-Shaw Flows

摘要

In this paper, we investigate some geometric properties of the moving frontier of a viscous fluid for planar flows in Hele-Shaw cells under injection. We study invariance in time properties of the free boundary for bounded and unbounded domains (with bounded complement) under the assumption of zero surface tension. By applying certain results in the theory of univalent functions we partially solve an open problem of Vasil'ev concerning the invariance in time of starlikeness of order alpha, alpha is an element of[0,1), for bounded domains. In the case of unbounded domains with bounded complement we analyze the invariance in time of convexity of order alpha, alpha is an element of[0, 1).