  We present a variational approach for the construction of Leray-Hopf
solutions to the non-newtonian Navier-Stokes system. Inspired by the work
[OSS18] on the corresponding Newtonian problem, we minimise certain stabilised
Weighted Inertia-Dissipation-Energy (WIDE) functionals and pass to the limit of
a vanishing parameter in order to recover a Leray-Hopf solution of the
non-newtonian Navier-Stokes equations. It turns out that the results differ
depending on the rheology of the fluid. The investigation of the non-newtonian
Navier-Stokes system via this variational approach is motivated by the fact
that it is particularly well suited to gain insights into weak, respectively
strong convergence properties for different flow-behaviour exponents and thus
into possibly turbulent behaviour of the fluid flow. With this analysis we
extend the results of [BS22] to power-law exponents $\tfrac{2d}{d+2} < p <
\tfrac{3d+2}{d+2}$, where weak solutions do not satisfy the energy equality and
the involved convergence is genuinely weak. Key of the argument is to pass to
the limit in the nonlinear viscosity term in the time-dependent setting. For
this we provide an elliptic-parabolic solenoidal Lipschitz truncation that
might be of independent interest.
