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We provide existence of very weak solutions and new a-priori estimates forsteady flows of non-Newtonian fluids when the right-hand sides are not in thenatural existence class. To obtain the a-priori estimates we make use of anewly developed solenoidal Lipschitz truncation that preserves zero boundaryvalues. We provide also estimates in (Muckenhoupt) weighted spaces which permitus to regain a duality pairing. Our estimates are valid even in the presence ofthe convective term. They are obtained via a newly developed comparison methodthat allows to "cut out" the singularities of the right hand side such that theskew symmetry of the convective term can be used for large parts of the righthand side.

Existence of steady very weak solutions to Navier-Stokes equations with non-Newtonian stress tensors