Very weak solutions to non-Newtonian fluids

I will present a new result which was established in collaboration with M. Bulıcek and J. Burczak. We established an existence, uniqueness and optimal regularity results for very weak solutions to certain incompressible non-Newtonian fluids. We introduce structural assumptions of Uhlenbeck type on the stress tensor. These as-sumptions are sufficient and to some extend also necessary to built a unified theory. Our approach leads qualitatively to the same so called Lp-theory as the one that is available for the linear Stokes equation.