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