Density-Dependent Incompressible Fluids with Non-Newtonian Viscosity

Abstract

We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of p-coercivity and (p−1)-growth, for a given parameter p > 1. The existence of Dirichlet weak solutions was obtained in [2], in the cases p ≥ 12/5 if d = 3 or p ≥ 2 if d = 2, d being the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all p ≥ 2. In addition, we obtain regularity properties of weak solutions whenever p ≥ 20/9 (if d = 3) or p ≥ 2 (if d = 2). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained.