  Ericksen and Leslie established a theory to model the flow of nematic liquid
crystals. This paper is devoted to the Cauchy Problem of a simplified version
of their system, which retains most of the properties of the original one. We
consider the density-dependent case and we establish the global existence of
solutions in the whole space for small initial data. The initial density only
has to be bounded and kept far from vacuum, while the initial velocity belongs
to some critical Besov Space. Under a little bit more regularity for the
initial velocity, we prove also that those solutions are unique.
