| Title:
|
Density-dependent incompressible fluids with non-Newtonian viscosity (English) |
| Author:
|
Guillén-González, F. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
54 |
| Issue:
|
3 |
| Year:
|
2004 |
| Pages:
|
637-656 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
variable density |
| Keyword:
|
shear-dependent viscosity |
| Keyword:
|
power law |
| Keyword:
|
Carreau’s laws |
| Keyword:
|
weak solution |
| Keyword:
|
strong solution |
| Keyword:
|
periodic boundary conditions |
| MSC:
|
35B10 |
| MSC:
|
35D05 |
| MSC:
|
35M10 |
| MSC:
|
35Q35 |
| MSC:
|
76A05 |
| MSC:
|
76D03 |
| idZBL:
|
Zbl 1080.35004 |
| idMR:
|
MR2086722 |
| . |
| Date available:
|
2009-09-24T11:16:07Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127917 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
[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 |
| Reference:
|
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| Reference:
|
[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 |
| Reference:
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| . |
