Strong solutions for non-newtonian fluids for small times

In this talk we will consider the incompressible Navier-Stokes equations, where the extra stress is replaced by an operator which is induced by a suitable potential with $p$-growth. Hereby we restrict ourselves to the three dimensional space periodic case. First of all we will consider a small trick to increase the regularity of the velocity to $L^\infty(I, W^{1,6(p-1)}(\Omega))$. With this gained regularity in mind we will attack the problem of existence of strong solutions (for small times) for small values of $p$, namely $p >7/5$.