On the Existence, Uniqueness and $C^{1,\gamma} (\bar{\Omega}) \cap W^{2,2}(\Omega)$ Regularity for a Class of Shear-Thinning Fluids

Abstract.

We consider a stationary Navier–Stokes system with shear dependent viscosity, under Dirichlet boundary conditions. We prove Hölder continuity, up to the boundary, for the gradient of the velocity field together with the L ²-summability of the weak second derivatives. The results hold under suitable smallness assumptions on the force term and without any restriction on the range of $p \in (1, 2)$.