We first introduce various non-Newtonian fluids, and its motivation.
We then address the existence of strong solutions to
a system of equations of
motion of an incompressible non-Newtonian fluid.
Our aim is to prove the short-time existence of
strong solutions for the case of shear thickening viscosity,
which corresponds to the power law
nu(mathbfD)=|mathbfD| q2   $$extract_itex$$nu (mathbf D)= |mathbf D|^{q-2}$$/extract_itex$$ (2<q<+infty)  $$extract_itex$$(2< q<+infty )$$/extract_itex$$ .
In particular, we find that global strong
solutions exist whenever q>2.23cdots  $$extract_itex$$q > 2.23 cdots$$/extract_itex$$ .
The results are obtained by flattening the boundary
and by using the difference quotient method.
Near the boundary, we use weighted estimates in the normal direction.