A variational approach to the Navier-Stokes equations with shear-dependent viscosity

Abstract

We present a variational approach for the construction of Leray-Hopf solutions to the non-Newtonian Navier-Stokes system. Inspired by the work [42] 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. The investigation of the non-Newtonian Navier-Stokes system via this variational approach is particularly well suited to gain insights into weak, respectively strong convergence properties of approximating sequences for different flow-behaviour exponents. With this analysis we extend the results of [4] 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.