Local-in-time well-posedness of the initial-boundary value problem for a class of non-Newtonian Navier-Stokes problems on domains with compact C 3 -boundary is proven in an Lp-setting for any space dimension n > 2. The stress tensor is assumed to be of the generalized Newtonian type, i.e., S = 2μ(|e| 2  2 )e - πI, £ = 1 2(∇u + ∇u T ), where |e| 2  2  = Σ n  i,j=1  e 2  ij  denotes the Hilbert-Schmidt norm of the rate of strain tensor e. The viscosity function μ ∈ C 2-  (R + ) is subject only to the condition μ(s) > 0, μ(s) + 2sμ'(s) > 0, s > 0, which for the standard power-law-like function μ(s) = μ 0 (1+ s) d-2 2 merely means μ 0  > 0 and d > 1. This result is based on maximal regularity theory for a suitable linear problem and a contraction argument.
['Dieter Bothe', 'Jan Prüss']
Lp-THEORY FOR A CLASS OF NON-NEWTONIAN FLUIDS