Local and global existence for the Ericksen - Leslie problem in unbounded domains

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the space R^N with N≥3, on R₊³ and on Ω⊆R³ exterior domain with sufficiently smooth boundary. The crystal orientation is described by unit vector v that is a small perturbation of a fixed constant vector η. We prove through a combination of energy method with dispersive a priori estimates a local existence and global existence for small initial data by a contraction argument. In particular, we obtain the following regularity of the liquid velocity u and of the crystal orientation v u∈L^∞((0,T);H^s(Ω)),∇u∈L²((0,T);H^s(Ω)), ∇v∈L^∞((0,T);H^s(Ω)),∇²v∈L²((0,T);H^s(Ω)) for [Formula presented] if Ω=R^N and [Formula presented] if Ω=R₊³ or in the exterior case, asking low regularity assumptions on u₀ and v₀.