Preprint D34/2006
NAVIER-STOKES EQUATIONS WITH SHEAR DEPENDENT VISCOSITY. REGULARITY UP TO THE BOUNDARY.

Hugo Beirao da Veiga

**Keywords: **
Generalized Navier-Stokes equations | Non-Newtonian fluids | Regularity up to the boundary |

We prove sharp regularity results for the stationary and the evolution Navier-Stokes equations with shear dependent viscosity, see equation (1.1), under the non-slip boundary condition (1.4). We are interested in regularity results up to the boundary for all the second order derivatives of the velocity and all the first order derivatives of the pressure, in dimension 3. We consider a cubic domain and impose our boundary condition (1.4) only on two opposite faces. On the other faces we assume periodicity, as a device to avoid effective boundary conditions. This choice is made so that we work in a bounded domain $\Omega$ and simultaneously with a flat boundary. In the last section we provide the extension of the results from the stationary to the evolution problem. In a forthcoming paper we extend the results to arbitrary, regular, open sets.