# Article

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Keywords:
variable density; shear-dependent viscosity; power law; Carreau’s laws; weak solution; strong solution; periodic boundary conditions
Summary:
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.
References:
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