Time Periodic Solutions to the Evolutionary Oseen Model for a Generalized Newtonian Incompressible Fluid

Abstract

In this paper we study a nonstationary Oseen model for a generalized Newtonian incompressible fluid with a time periodic condition and a multivalued, nonmonotone friction law. First, a variational formulation of the model is obtained; that is a nonlinear boundary hemivariational inequality of parabolic type for the velocity field. Then, an abstract first-order evolutionary hemivariational inequality in the framework of an evolution triple of spaces is investigated. Under mild assumptions, the nonemptiness and weak compactness of the set of periodic solutions to the abstract inequality are proven. Furthermore, a uniqueness theorem for the abstract inequality is established by using a monotonicity argument. Finally, we employ the theoretical results to examine the nonstationary Oseen model.