Regularity of Non-Newtonian Fluids

Abstract

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).