We consider steady flows of shear thinning fluids in bounded domains under the action of external forces and Dirichlet boundary conditions. For a power-law index $q \in (2d/(d+2),3d/(d+2)]$, we construct weak solutions to the nonhomogeneous boundary value problem assuming that the boundary data is small enough.
Moreover, under the restriction $q \in ((2d-1)/d,2)$, $d=2,3$, and extra regularity for the boundary data, we construct weak solutions by extending the tangential part of the velocity at the boundary in such a way that it is possible to partially control the inertial term. This imposes restrictions only on the size of the normal component of the boundary data.

CEMAT - Center for Computational and Stochastic Mathematics