J\"org Wolf
Interior regularity of weak solutions to the equations of a stationary motion of a non-Newtonian fluid with shear-dependent viscosity. The case $q=\frac {3d}{d+2}$

Comment.Math.Univ.Carolin. 48,4 (2007) 659-668.

Abstract: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 $u$.

Keywords: non-Newtonian fluids, weak solutions, interior regularity
AMS Subject Classification: 35Q30, 35B65, 76A05