Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids

Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalized Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $\Omega \subset \mathbb{R}^d$, $d \in \{2,3\}$, we investigate a fully-discrete approximation scheme, using a spatial mixed finite element approximation combined with backward Euler time-stepping. We show convergence of a subsequence of approximate solutions, when the velocity field belongs to the space of solenoidal functions contained in $L^\infty(0,T;L^2(\Omega)^d)\cap L^q(0,T;W^{1,q}_0(\Omega)^d)$, provided that $q\in \big(\frac{2d}{d+2},\infty\big)$, which is the maximal range for $q$ with respect to existence of weak solutions. This is achieved by a technique based on splitting and regularizing, the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness results.