  The work deals with the Ericksen-Leslie model for nematic liquid crystals on
the whole space, the half-space and on exterior domains with smooth boundary.
The crystal orientation is described by a unit vector that is a small
perturbation of a fixed constant vector. We prove, through a combination of
energy method with dispersive a priori estimates, a local existence and a
global existence for small initial data by a contraction argument asking low
regularity assumptions on the initial condition.
