  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 with the basis
constituted by the eigenfunctions of the periodic Bessel-potential operator, we
prove the existence of a global weak solution.
