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 RN 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);Hs(Ω)),∇u∈L2((0,T);Hs(Ω)), ∇v∈L∞((0,T);Hs(Ω)),∇2v∈L2((0,T);Hs(Ω)) for [Formula presented] if Ω=RN and [Formula presented] if Ω=R+3 or in the exterior case, asking low regularity assumptions on u0 and v0.