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Keywords:
non-Newtonian fluids; weak solutions; interior regularity
non-Newtonian fluids; weak solutions; interior regularity
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 $u$.
References:
[1] Adams R.A.: Sobolev Spaces. Academic Press, Boston, 1978. Zbl 1098.46001
[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. $2^{an{nd}}$ 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]. MR 1602949
[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
[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
[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. MR 0717034 | Zbl 0516.49003
[9] Giaquinta M., Modica G.: Almost-everywhere regularity results for solutions of nonlinear elliptic systems. Manuscripta Math. 28 (1979), 109-158. MR 0535699 | Zbl 0411.35018
[10] Lamb H.: Hydrodynamics. $6^{an{th}}$ ed., Cambridge Univ. Press, Cambridge, 1945. Zbl 0828.01012
[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. MR 2177130 | Zbl 1070.35023
[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
[13] Wilkinson W.L.: Non-Newtonian Fluids. Fluid Mechanics, Mixing and Heat Transfer. Pergamon Press, London, New York, 1960. MR 0110392 | Zbl 0124.41802
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