We deal with a system of partial diﬀerential equations describing a steady ﬂow of a homogeneous incompressible non-Newtonian ﬂuid with pressure and shear rate dependent viscosity subject to the homogeneous Dirichlet (no-slip) boundary condition. We establish a global existence of a weak solution for a certain class of such ﬂuids in which the dependence of the viscosity on the shear rate is polynomiallike, characterized by the power-law index. A decomposition of the pressure and Lipschitz approximations of Sobolev functions are considered in order to obtain almost everywhere convergence of the pressure and the symmetric part of the velocity gradient and thus obtain new existence results for low value of the power-law index.
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Miroslav Bulíček, V. Fišerová, Existence Theory for Steady Flows of Fluids with Pressure and Shear Rate Dependent Viscosity, for Low Values of the Power-Law Index. Z. Anal. Anwend. 28 (2009), no. 3, pp. 349–371DOI 10.4171/ZAA/1389