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Density-dependent incompressible fluids with non-Newtonian viscosity. (English) Zbl 1080.35004

Summary: We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and nonconstant 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 E.Fernández-Cara, F.Guillén and R.R.Ortega [Nonlinear Anal., Theory Methods Appl.28, 1079-1100 (1997; Zbl 0865.35103)], in the cases \(p\geq 12/5\) if \(d=3\) or \(p \geq 2\) if \(d=2\), \(d\) being the dimension of the domain. With help of some new estimates (which lead to pointwise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all \(p \geq 2\). In addition, we obtain regularity properties of weak solutions whenever \(p \geq 20/9\) (if \(d = 3\)) or \(p \geq 2\) (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.

MSC:

35B10 Periodic solutions to PDEs
35M10 PDEs of mixed type
76A05 Non-Newtonian fluids

Citations:

Zbl 0865.35103
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References:

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