J. math. fluid mech. 7 (2005) 298–313 1422-6928/05/020298-16 c  2005 Birkh¨auser Verlag, Basel DOI 10.1007/s00021-004-0120-z Journal of Mathematical Fluid Mechanics Interior Differentiability of Weak Solutions to the Equations of Stationary Motion of a Class of Non-Newtonian Fluids J. Naumann and J. Wolf Communicated by G. P. Galdi Abstract. In this paper, we consider weak solutions to the equations of stationary motion of a class of non-Newtonian fluids the constitutive law of which includes the “power law model” as special case. We prove the existence of second order derivatives of weak solutions to these equations. Mathematics Subject Classification (2000). 35Q30, 76D05, 35J65. Keywords. Non-Newtonian fluids, difference estimates of weak solutions, fractional differentia- bility. 1. Introduction Equations and constitutive law. The stationary motion of an incompressible fluid through a bounded domain Ω ⊂ R d (d = 2 resp. d = 3) is governed by the conservation of momentum and the conservation of mass: − div(S − pI )+ u·∇u = div f in Ω, (1.1) div u =0 in Ω, (1.2) where S = {S ij } = deviatoric stress tensor, 1) p = pressure, I = unit tensor (S − pI = full stress tensor), u = {u 1 ,...,u d } = velocity, f = {f ij } = external force. 1) Throughout Latin subscripts take the values 1, 2 (if d = 2) or 1, 2, 3 (if d = 3). Repeated subscripts imply summation over 1, 2 or 1, 2, 3, respectively.