Geodesic
Density-dependent incompressible fluids with non-Newtonian viscosity
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 637-656 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 35B10, 35D05, 35M10, 35Q35, 76A05, 76D03
Keywords: variable density; shear-dependent viscosity; power law; Carreau’s laws; weak solution; strong solution; periodic boundary conditions 
@article{CMJ_2004_54_3_a7,
     author = {Guill\'en-Gonz\'alez, F.},
     title = {Density-dependent incompressible fluids with {non-Newtonian} viscosity},
     journal = {Czechoslovak Mathematical Journal},
     pages = {637--656},
     year = {2004},
     volume = {54},
     number = {3},
     mrnumber = {2086722},
     zbl = {1080.35004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a7/}
}
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Guillén-González, F. Density-dependent incompressible fluids with non-Newtonian viscosity. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 637-656. http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a7/