On flows of an incompressible fluid with a discontinuous power-law-like rheology

We establish the mathematical theory for the steady and unsteady flows of fluids with discontinuous constitutive equations. We consider a model for a fluid that at certain critical value of the shear rate exhibits jumps in the generalized viscosity of a power-law type. Using tools such as Young measures, maximal monotone operators, compact embeddings and energy equality we prove the existence of a solution to the problem in consideration. In this approach, Galerkin approximations converge strongly to the solution of the original problem.