LP-theory for a class of non-newtonian fluids

Local-in-time well-posedness of the initial-boundary value problem for a class of non-Newtonian Navier-Stokes problems on domains with compact C3-- boundary is proven in an L-p-setting for any space dimension n >= 2. The stress tensor is assumed to be of the generalized Newtonian type, i.e., S = 2 mu(vertical bar epsilon vertical bar(2)(2))epsilon - pi I, epsilon = 1/2 (del u + del u(T)), where vertical bar epsilon vertical bar(2)(2) = Sigma(n)(i, j=1) epsilon(2)(ij) denotes the Hilbert - Schmidt norm of the rate of strain tensor epsilon. The viscosity function mu is an element of C2-(R+) is subject only to the condition mu(s) > 0, mu(s) + 2s mu'(s) > 0, s >= 0, which for the standard power-law-like function mu(s) = mu(0)(1 + s) (d-2/2) merely means mu(0) > 0 and d >= 1. This result is based on maximal regularity theory for a suitable linear problem and a contraction argument.