On Analysis of Steady Flows of Fluids with Shear-Dependent Viscosity Based on the Lipschitz Truncation Method

Abstract

We deal with a system of partial differential equations describing a steady motion of an incompressible fluid with shear-dependent viscosity and present a new global existence result for $ p>\frac{2d}{d+2} $. Here p is the coercivity parameter of the nonlinear elliptic operator related to the stress tensor and d is the dimension of the space. Lipschitz test functions, a subtle splitting of the level sets of the maximal functions for the velocity gradients, and a decomposition of the pressure are incorporated to obtain almost everywhere convergence of the velocity gradients.