On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case $p\geq2$

Abstract

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 $\mathbb R ^3$ with Dirichlet boundary conditions. The nonlinear elliptic operator, which is related to the stress tensor, has $p$ structure. The paper deals with the case $p\ge 2$, for which the existence of weak solutions is proved. If $p\ge \frac{9}{4}$ then a weak solution is strong and unique among all weak solutions.