Attractors for Navier–Stokes flows with multivalued and nonmonotone subdifferential boundary conditions

We consider two-dimensional nonstationary Navier–Stokes shear flow with multivalued and nonmonotone boundary conditions on a part of the boundary of the flow domain. We prove the existence of global in time solutions of the considered problem which is governed by a partial differential inclusion with a multivalued term in the form of Clarke subdifferential. Then we prove the existence of a trajectory attractor and a weak global attractor for the associated multivalued semiflow.This research is motivated by control problems for fluid flows in domains with semipermeable walls and membranes.