Abstract
We consider a double-phase non-Newtonian fluid, described by a stress tensor which is the sum of a
-Stokes and a
-Stokes stress tensor, with 1 <
<2 <
<∞. For a wide range of parameters (
,
), we prove the uniqueness of small solutions. We use the
< 2 features to obtain quadratic-type estimates for the stress-tensor, while we use the improved regularity coming from the term with
> 2 to justify calculations for weak solutions. Results are obtained through a careful use of the symmetries of the convective term and are also valid for rather general (even anisotropic) stress-tensors.