Flow of Dilute Aqueous Polymer Solutions in a Heterogeneous Porous Medium: Existence Results for Steady and Unsteady Cases

In this paper, we consider a mathematical model for the flow of a dilute aqueous polymer solution through a heterogeneous porous medium. On the boundary of the flow region, the impermeability condition and a nonlinear Navier-type slip condition are prescribed. Our goal is to investigate the existence of weak solutions to the governing equations of this model, which is a challenging problem for both steady and unsteady flows. To prove the weak solvability, we use a modified Galerkin scheme with special basis elements in appropriate Sobolev spaces. We obtain the existence results without assuming smallness of the model data for the boundary value problem related to steady flows and prove the global-in-time solvability of the initial-boundary value problem describing unsteady flows. Moreover, energy equalities are established for weak solutions possessing additional regularity. Our results can serve as a starting point for further research on the problems under consideration, including numerical analysis and optimal control of polymer fluid flows.