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variable density; shear-dependent viscosity; power law; Carreau’s laws; weak solution; strong solution; periodic boundary conditions
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 $p \ge 12/5$ if $d = 3$ or $p \ge 2$ 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 $p \ge 2$. In addition, we obtain regularity properties of weak solutions whenever $p \ge 20/9$ (if $d = 3$) or $p \ge 2$ (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.
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