, Volume 7, Issue 3, pp 413-450
Date: 14 Jun 2005

Strong Solutions for Generalized Newtonian Fluids

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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 \(\tfrac{7}{5} < \inf p(t,x) \leq \sup p(t,x) \leq 2.\)