Analysis of generalized Navier-Stokes equations for stationary shear thickening flows

We discuss solutions u : R-3 superset of Omega -> R-3, pi : Omega -> R to generalized Navier-Stokes equations

div sigma = (del u)u + del pi - f,

sigma = sigma(epsilon(u)) = mu(vertical bar epsilon(u)vertical bar)epsilon(u),

with generalized viscosity function mu. Here u denotes the velocity field, pi the pressure, sigma the stress deviator and f an external volume force. Since we are interested in shear thickening flows mu is assumed to be increasing but we do not assume any growth condition. The result is the existence of a weak solution to the equation above with V(epsilon(u)) is an element of W-loc(1,2) (Omega), where V(epsilon) = integral(vertical bar epsilon vertical bar)(0) root mu(s) ds. Moreover, we have u is an element of C-1,C-alpha(Omega(0), R-3) for any alpha <1, where Omega(0) subset of Omega is an open set with dim(H)(Omega \ Omega(0))