Convergence to equilibrium for smectic-A liquid crystals in 3D domains without constraints for the viscosity

In this paper, we focus on a smectic-A liquid crystal model in 3D domains, and obtain three main results: the proof of an adequate Lojasiewicz-Simon inequality by using an abstract result, the rigorous proof (via a Galerkin approach) of the existence of global in-time weak solutions that become strong (and unique) in long-time, and its convergence to equilibrium of the whole trajectory as time goes to infinity. Given any regular initial data, the existence of a unique global in-time regular solution (bounded up to infinite time) and the convergence to an equilibrium have been previously proved under the constraint of a sufficiently high level of viscosity. Here, all results are obtained without imposing said constraint.