  Long-time and large-data existence of weak solutions for initial- and
boundary-value problems concerning three-dimensional flows of
\emph{incompressible} fluids is nowadays available not only for Navier--Stokes
fluids but also for various fluid models where the relation between the Cauchy
stress tensor and the symmetric part of the velocity gradient is
\emph{nonlinear}. The majority of such studies however concerns models where
such a dependence is \emph{explicit} (the stress is a function of the velocity
gradient), which makes the class of studied models unduly restrictive. The same
concerns boundary conditions, or more precisely the slipping mechanisms on the
boundary, where the no-slip is still the most preferred condition considered in
the literature. Our main objective is to develop a robust mathematical theory
for unsteady internal flows of \emph{implicitly constituted} incompressible
fluids with implicit relations between the tangential projections of the
velocity and the normal traction on the boundary. The theory covers numerous
rheological models used in chemistry, biorheology, polymer and food industry as
well as in geomechanics. It also includes, as special cases, nonlinear slip as
well as stick-slip boundary conditions. Unlike earlier studies, the conditions
characterizing admissible classes of constitutive equations are expressed by
means of tools of elementary calculus. In addition, a fully constructive proof
(approximation scheme) is incorporated. Finally, we focus on the question of
uniqueness of such weak solutions.
