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

Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data