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Weak solutions to stationary shear thinning fluids with nonhomogeneous Dirichlet boundary conditions

Elshehabey, Hillal M.; Silvestre, Ana Leonor

Nonlinear Anal. Theory Methods Appl., 208 (2021), 11227

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.