Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case q = 3 d d + 2

Jörg Wolf

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 4, page 659-668
  • ISSN: 0010-2628

Abstract

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In this paper we consider weak solutions 𝐮 : Ω → ℝ d to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain Ω ⊂ ℝ d ( d = 2 or d = 3 ). For the critical case q = 3 d d + 2 we prove the higher integrability of ∇ 𝐮 which forms the basis for applying the method of differences in order to get fractional differentiability of ∇ 𝐮 . From this we show the existence of second order weak derivatives of u .

How to cite

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Wolf, Jörg. "Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case $q=\frac{3d}{d+2}$." Commentationes Mathematicae Universitatis Carolinae 48.4 (2007): 659-668. <http://eudml.org/doc/250208>.

@article{Wolf2007,
abstract = {In this paper we consider weak solutions $\{\mathbf \{u\}\}: \Omega \rightarrow \mathbb \{R\}^d$ to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain $\Omega \subset \mathbb \{R\}^d$ ($d=2$ or $d=3$). For the critical case $q=\frac\{3d\}\{d+2\}$ we prove the higher integrability of $\nabla \{\mathbf \{u\}\}$ which forms the basis for applying the method of differences in order to get fractional differentiability of $\nabla \{\mathbf \{u\}\}$. From this we show the existence of second order weak derivatives of $u$.},
author = {Wolf, Jörg},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {non-Newtonian fluids; weak solutions; interior regularity; non-Newtonian fluids; weak solutions; interior regularity},
language = {eng},
number = {4},
pages = {659-668},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case $q=\frac\{3d\}\{d+2\}$},
url = {http://eudml.org/doc/250208},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Wolf, Jörg
TI - Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case $q=\frac{3d}{d+2}$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 4
SP - 659
EP - 668
AB - In this paper we consider weak solutions ${\mathbf {u}}: \Omega \rightarrow \mathbb {R}^d$ to the equations of stationary motion of a fluid with shear dependent viscosity in a bounded domain $\Omega \subset \mathbb {R}^d$ ($d=2$ or $d=3$). For the critical case $q=\frac{3d}{d+2}$ we prove the higher integrability of $\nabla {\mathbf {u}}$ which forms the basis for applying the method of differences in order to get fractional differentiability of $\nabla {\mathbf {u}}$. From this we show the existence of second order weak derivatives of $u$.
LA - eng
KW - non-Newtonian fluids; weak solutions; interior regularity; non-Newtonian fluids; weak solutions; interior regularity
UR - http://eudml.org/doc/250208
ER -

References

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  1. Adams R.A., Sobolev Spaces, Academic Press, Boston, 1978. Zbl1098.46001
  2. Astarita G., Marrucci G., Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, London, New York, 1974. 
  3. Batchelor G.K., An Introduction to Fluid Mechanics, Cambridge Univ. Press, Cambridge, 1967. 
  4. Bird R.B., Armstrong R.C., Hassager O., Dynamics of Polymer Liquids. Vol. 1: Fluid Mechanics, ed., J. Wiley Sons, New York, 1987. 
  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]. (1997) MR1602949
  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. (2004) MR2001659
  7. Galdi G.P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems, Springer, New York, 1994. MR1284205
  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. Zbl0516.49003MR0717034
  9. Giaquinta M., Modica G., Almost-everywhere regularity results for solutions of nonlinear elliptic systems, Manuscripta Math. 28 (1979), 109-158. (1979) Zbl0411.35018MR0535699
  10. Lamb H., Hydrodynamics, ed., Cambridge Univ. Press, Cambridge, 1945. Zbl0828.01012
  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. (2005) Zbl1070.35023MR2177130
  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]. (1997) MR1602945
  13. Wilkinson W.L., Non-Newtonian Fluids. Fluid Mechanics, Mixing and Heat Transfer, Pergamon Press, London, New York, 1960. Zbl0124.41802MR0110392

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