Existence for a Class of Non-Newtonian Fluids with a Non-local Friction Boundary Condition

We deal with a variational inequality describing the motion of incompressible fluids, whose viscous stress tensors belong to the subdifferential of a functional at the point given by the symmetric part of the velocity gradient, with a nonlocal friction condition on a part of the boundary obtained by a generalized mollification of the stresses. We establish an existence result of a solution to the nonlocal friction problem for this class of non.Newtonian flows. The result is based on the Faedo.Galerkin and Moreau.Yosida methods, the duality theory of convex analysis and the Tychonov.Kakutani.Glicksberg fixed point theorem for multi.valued mappings in an appropriate functional space framework.