We consider the equations of a non-Newtonian incompressible fluid in a general time space cylinder $Q_{T}= \Omega \times (0,T) \subset \mathbb{R}^{n} \times \mathbb{R}, n \geq 2$. We assume that the rheology of the fluid is changing with respect to time and space and satisfies for each $(x,t) \in Q_{T}$ the associated power law $ |D|^{p(x,t) } D $. Under the assumption that $ \frac{2n}{n+2} \lt p_{0} \le p(x,t) \leq p_{1} \lt +\infty$ and the set of discontinuity of $p$ is closed and of measure zero we show the existence of a weak solution to the corresponding equations of PDEs for any given initial velocity in $L^{2}_{\sigma } (\Omega) $. Joint work with Prof. H-O. Bae (Ajou University, Suwon).