Strong Solutions for Generalized Newtonian Fluids

Abstract.

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 $p > \tfrac{5}{3}$ (see [16]) to $p > \tfrac{7}{5}.$ Moreover, for $\tfrac{7}{5} < p \leq 2$ we improve the regularity of the velocity field and show that ${\mathbf{u}} \in C([0,T],W_{{\text{div}}}^{1,6(p - 1) - \epsilon } (\Omega ))$ for all $\epsilon > 0.$ Within this class of regularity, we prove uniqueness for all $p > \tfrac{7}{5}.$ 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 $\tfrac{7}{5} < \inf p(t,x) \leq \sup p(t,x) \leq 2.$