  We consider evolution (non-stationary) space-periodic solutions to the
$n$-dimensional non-linear Navier-Stokes equations of anisotropic fluids with
the viscosity coefficient tensor variable in space and time and satisfying the
relaxed ellipticity condition. Employing the Galerkin algorithm, we prove the
existence of Serrin-type solutions, that is, weak solutions with the velocity
in the periodic space $L_2(0,T;\dot{\mathbf H}^{n/2}_{\#\sigma})$, $n\ge 2$.
The solution uniqueness and regularity results are also discussed.
