  In this paper, we prove the local well-posedness of 3-D density-dependent
liquid crystal flows with initial data in the critical Besov spaces, without
assumptions of small density variation. Furthermore, if the initial density is
close enough to a positive constant and the critical Besov norm of the liquid
crystal orientation field and the horizontal components of the initial velocity
field polynomially small compared with the critical Besov norm to the veritcal
component of the initial velocity field, then the system has a unique global
solution.
