J. math. fluid mech. 11 (2009) 171–185 1422-6928/09/020171-15 c  2008 Birkh¨ auser Verlag, Basel DOI 10.1007/s00021-008-0254-5 Journal of Mathematical Fluid Mechanics On the W 2,q -Regularity of Incompressible Fluids with Shear- Dependent Viscosities: The Shear-Thinning Case Luigi C. Berselli Communicated by H. Beir˜ ao da Veiga Abstract. In this paper we improve the results stated in Reference [2], in this same Journal, by using -basically- the same tools. We consider a non Newtonian fluid governed by equations with p-structure and we show that second order derivatives of the velocity and first order derivatives of the pressure belong to suitable Lebesgue spaces. Mathematics Subject Classification (2000). Primary 76A05; secondary 35B65, 35Q35. Keywords. Shear dependent viscosity, incompressible fluid, regularity up to the boundary. 1. Introduction and main results In this paper we consider an incompressible fluid, with viscosity ν = ν (Du) de- pending on a power (power-law fluid of p-fluid) of the deformation tensor Du = 1 2 [∇u +(∇u) T ]: ν (Du)= ν 0 (µ + |Du|) p−2 with ν 0 ,µ> 0, p> 1. (1.1) We study the following system of partial differential equations  −∇ ·  (µ + |Du|) p−2 Du  +(u ·∇) u + ∇π = f in Ω, ∇· u =0 in Ω, (1.2) in the case of shear-thinning, i.e., we assume 1 <p< 2 and – without loss of generality – we assume ν 0 = 1. We consider the problem in a flat domain by assuming zero Dirichlet data on the boundary. Beir˜ ao da Veiga, in the paper [2], considers the Dirichlet problem in a domain with a flat boundary (see below). Here we show how to improve the results in [2] by using anisotropic Sobolev-type inequalities and by resorting to additional features of the structure of the equations. In order to make the paper easier to read, we try to be self-contained and the missing details can be found in [2]. In particular, we