Generalized Darcy–Oseen resolvent problem

In this paper we study the well-posedness of a coupled Darcy-Oseen resolvent problem, describing the fluid flow between free fluid domains and porous media separated by a semipermeable membrane. The influence of osmotic effects, induced by the presence of a semipermeable membrane, on the flow velocity is reflected in the transmission conditions on the surface between the free-fluid domain and the porous medium. To prove the existence of a weak solution of the generalized Darcy-Oseen resolvent system we consider two auxiliary problem: a mixed Navier-Dirichlet problem for the generalized Oseen resolvent system and Robin problem for an elliptic equation related to the general Darcy equations