Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows

We analyze a simplified Ericksen-Leslie model for nematic liquid crystal flows firstly introduced in [18] with non-autonomous forcing bulk term and boundary conditions on the order parameter field. We obtain existence of weak solutions in the two- and three-dimensional cases. We prove uniqueness, continuous dependence on initial conditions, forcing and boundary terms and also existence of strong solutions in the 2D case. Focusing on the 2D case, we then study the long term behavior of solutions by obtaining existence of global attractors for normal forcing terms (according to [21]). Finally, we prove the existence of exponential attractors for quasi-periodic forcing terms in the 2D model.