On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p≥2

Existence and regularity properties of solutions for the evolutionary system describing unsteady flows of incompressible fluids with shear dependent viscosity are studied. The problem is considered in a bounded, smooth domain of R ³ with Dirichlet boundary conditions. The nonlinear elliptic operator, which is related to the stress tensor, has p structure. The paper deals with the case p≥ 2, for which the existence of weak solutions is proved. If p≥ 9/4 then a weak solution is strong and unique among all weak solutions.