Regularity of Non-Newtonian Fluids

In this paper, we consider a non-Newtonian fluids with shear dependent viscosity in a bounded domain ${Omega subset mathbb{R}^n, n = 2, 3}$ . For the power-law model with the viscosity as in (1.4), we show the global in time existence of a weak solution for ${q geq frac{11}{5}}$ when n = 3 (see Theorem 1.1), and the local in time existence of a weak solution for ${2 > q > frac{3n}{n+2}}$ , when n = 2,3 (see Theorem 1.2).