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| In our talk we discuss the local regularity properties of weak solutions to the equations of unsteady motions of a non-Newtonian fluid with shear rate dependent viscosity $\nu\sim |{\boldsymbol D}({\boldsymbol u})|^ {q-2}$ ($q>0$). We consider weak solutions
${\boldsymbol u} \in C_{w} ([0,T]; {\boldsymbol L}^2_\sigma (\Omega ))$ with ${\boldsymbol D}( {\boldsymbol u} )\in {\boldsymbol L}^q(Q_T)$ satisfying the corresponding integral identity for all smooth solenoidal test functions. For such weak solutions
we present a sufficient condition on ${\boldsymbol f}$ which implies that ${\boldsymbol u}$ is continuous in $Q$. The proof of this result is divided into three steps. First, based on the local pressure decomposition we prove the weak differentiability of $ {\boldsymbol V}= |{\boldsymbol D}({\boldsymbol u})|^{(q-2)/2} {\boldsymbol D}({\boldsymbol u}) $. Secondly, by using the local pressure decomposition $p= \frac {\partial p_H} {\partial t}+ p_0$ we verify the time regularity $\frac {\partial } {\partial t}({\boldsymbol u} +\nabla p_H) \in L^2(0,T; {\boldsymbol L}^2_{\rm loc} (\Omega ))$. Finally, with help of the method of differences with respect to time together with a standard scaling argument we obtain the required regularity property of ${\boldsymbol u}$. |
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