  We prove short-time well-posedness and existence of global weak solutions of
the Beris--Edwards model for nematic liquid crystals in the case of a bounded
domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The
system consists of the Navier-Stokes equations coupled with an evolution
equation for the $Q$-tensor. The solutions possess higher regularity in time of
order one compared to the class of weak solutions with finite energy. This
regularity is enough to obtain Lipschitz continuity of the non-linear terms in
the corresponding function spaces. Therefore the well-posedness is shown with
the aid of the contraction mapping principle using that the linearized system
is an isomorphism between the associated function spaces.
