Stability of the equilibria for periodic Stokesian Hele-Shaw flows

In this paper we consider the 2-dimensional flow of a Stokesian fluid in a Hele-Shaw cell. The motion of the flow is modelled by a modified Darcy's law. The existence of local solutions has been proved by the authors in a recent work, cf. [4]. The purpose of this paper is to identify the steady states of this flow and to study their stability. The equilibria will be identified as solutions of elliptic free boundary problems. It is shown that if the pressure on the bottom is constant then the corresponding steady state is asymptotically stable.