Some remarks on certain invariant geometric properties in Hele–Shaw flows

In this paper we study the time evolution of the free boundary of a viscous fluid for planar flows in Hele–Shaw cells under injection. We discuss the geometrical properties of the moving frontier for bounded and unbounded (with bounded complement) domains under the assumption of zero surface tension. Applying methods from the theory of univalent functions we prove the invariance in time of α-convexity. Moreover, we establish an upper bound for the order of strongly starlikeness of the classical solution in the Hele–Shaw problem which starts with a starlike bounded domain.