Strong solutions for generalized Newtonian fluids

We consider the motion of a generalized Newtonian fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). We focus on the three dimensional case with periodic boundary conditions and extend the existence result for strong solutions for small times from p > 5/3 (see [16]) to p > 7/5. Moreover, for 7/5 < p <= 2 we improve the regularity of the velocity field and show that u is an element of C([0, T], W-div(1,6(p- 1)-epsilon) ( Omega)) for all epsilon > 0. Within this class of regularity, we prove uniqueness for all p > 7/5. We generalize these results to the case when p is space and time dependent and to the system governing the flow of electrorheological fluids as long as 7/5 < inf p(t, x) <= sup p(t, x) <= 2.