Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
variable density; shear-dependent viscosity; power law; Carreau’s laws; weak solution; strong solution; periodic boundary conditions
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:
[1] S. A. Antonzev and A. V. Kazhikhov: Mathematical Study of Flows of Non-Homogeneous Fluids. Lectures at the University of Novosibirsk, U.S.S.R, 1973. (Russian)
[2] E. Fernández-Cara, F. Guillén and R. R. Ortega: Some theoretical results for visco-plastic and dilatant fluids with variable density. Nonlinear Anal. 28 (1997), 1079–1100. DOI 10.1016/S0362-546X(97)82861-1 | MR 1422802
[3] J. Frehse, J. Málek and M. Steinhauer: On existence results for fluids with shear dependent viscosity—unsteady flows. Partial Differential Equations, Praha 1998 Chapman & Hall/CRC, Res. Notes Math., 406, Boca Raton, FL, 2000, pp. 121–129. MR 1713880
[4] A. V. Kazhikhov: Resolution of boundary value problems for nonhomogeneous viscous fluids. Dokl. Akad. Nauk 216 (1974), 1008–1010. MR 0430562
[5] O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, 1969. MR 0254401 | Zbl 0184.52603
[6] J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, 1969. MR 0259693 | Zbl 0189.40603
[7] P. L. Lions: Mathematical Topics in Fluid Mechanics. Volume 1, Incompressible models. Clarendon Press, 1996. MR 1422251
[8] J. Málek, K. R. Rajagopal and M. Růžička: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models and Methods in Applied Sciences 5 (1995), 789–812. MR 1348587
[9] J. Málek, J. Nečas, M. Rokyta and M. Růžička: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, 1996. MR 1409366
[10] J. Málek, J. Nečas and M. Růžička: On weak solutions of non-Newtonian incompressible fluids in bounded three-dimensional domains. The case $p \ge 2$. Advances in Differential Equations 6 (2001), 257–302. MR 1799487
[11] J. Simon: Compact sets in $L^p(0,T;B)$. Ann. Mat. Pura Appl. 4 (1987), 65–96. MR 0916688
Search
Partner of
