  The present contribution investigates the well-posedness of a PDE system
describing the evolution of a nematic liquid crystal flow under kinematic
transports for molecules of different shapes. More in particular, the evolution
of the {\em velocity field} $\ub$ is ruled by the Navier-Stokes incompressible
system with a stress tensor exhibiting a special coupling between the transport
and the induced terms. The dynamic of the {\em director field} $\bd$ is
described by a variation of a parabolic Ginzburg-Landau equation with a
suitable penalization of the physical constraint $|\bd|=1$. Such equation
accounts for both the kinematic transport by the flow field and the internal
relaxation due to the elastic energy. The main aim of this contribution is to
overcome the lack of a maximum principle for the director equation and prove
(without any restriction on the data and on the physical constants of the
problem) the existence of global in time weak solutions under physically
meaningful boundary conditions on $\bd$ and $\ub$.
