| Title:
|
Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case $q=\frac{3d}{d+2}$ (English) |
| Author:
|
Wolf, Jörg |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
4 |
| Year:
|
2007 |
| Pages:
|
659-668 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider weak solutions ${\bold u}: \Omega \rightarrow \Bbb R^d$ to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain $\Omega \subset \Bbb R^d$ ($d=2$ or $d=3$). For the critical case $q=\frac{3d}{d+2}$ we prove the higher integrability of $\nabla {\bold u}$ which forms the basis for applying the method of differences in order to get fractional differentiability of $\nabla {\bold u}$. From this we show the existence of second order weak derivatives of $u$. (English) |
| Keyword:
|
non-Newtonian fluids |
| Keyword:
|
weak solutions |
| Keyword:
|
interior regularity |
| MSC:
|
35B65 |
| MSC:
|
35D10 |
| MSC:
|
35D30 |
| MSC:
|
35Q30 |
| MSC:
|
35Q35 |
| MSC:
|
76A05 |
| idZBL:
|
Zbl 1199.35297 |
| idMR:
|
MR2375166 |
| . |
| Date available:
|
2009-05-05T17:05:28Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119688 |
| . |
| Reference:
|
[1] Adams R.A.: Sobolev Spaces.Academic Press, Boston, 1978. Zbl 1098.46001 |
| Reference:
|
[2] Astarita G., Marrucci G.: Principles of Non-Newtonian Fluid Mechanics.McGraw-Hill, London, New York, 1974. |
| Reference:
|
[3] Batchelor G.K.: An Introduction to Fluid Mechanics.Cambridge Univ. Press, Cambridge, 1967. |
| Reference:
|
[4] Bird R.B., Armstrong R.C., Hassager O.: Dynamics of Polymer Liquids. Vol. 1: Fluid Mechanics.$2^{an{nd}}$ ed., J. Wiley & Sons, New York, 1987. |
| Reference:
|
[5] Frehse J., Málek J., Steinhauer M.: An existence result for fluids with shear dependent viscosity-steady flows.Nonlinear Anal. 30 5 (1997), 3041-3049; [Proc. 2nd World Congress Nonlin. Analysts]. MR 1602949 |
| Reference:
|
[6] Frehse J., Málek J., Steinhauer M.: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method.SIAM J. Math. Anal. 34 5 (2004), 1064-1083. MR 2001659 |
| Reference:
|
[7] Galdi G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems.Springer, New York, 1994. MR 1284205 |
| Reference:
|
[8] Giaquinta M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems.Annals Math. Studies, no. 105, Princeton Univ. Press, Princeton, N.J., 1983. Zbl 0516.49003, MR 0717034 |
| Reference:
|
[9] Giaquinta M., Modica G.: Almost-everywhere regularity results for solutions of nonlinear elliptic systems.Manuscripta Math. 28 (1979), 109-158. Zbl 0411.35018, MR 0535699 |
| Reference:
|
[10] Lamb H.: Hydrodynamics.$6^{an{th}}$ ed., Cambridge Univ. Press, Cambridge, 1945. Zbl 0828.01012 |
| Reference:
|
[11] Naumann J., Wolf J.: Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids.J. Math. Fluid Mech. 7 2 (2005), 298-313. Zbl 1070.35023, MR 2177130 |
| Reference:
|
[12] Růžička M.: A note on steady flow of fluids with shear dependent viscosity.Nonlinear Anal. 30 5 (1997), 3029-3039; [Proc. 2nd World Congress Nonlin. Analysts]. MR 1602945 |
| Reference:
|
[13] Wilkinson W.L.: Non-Newtonian Fluids. Fluid Mechanics, Mixing and Heat Transfer.Pergamon Press, London, New York, 1960. Zbl 0124.41802, MR 0110392 |
| . |
