Global well-posedness for the 3-D density-dependent liquid crystal flows in the critical Besov spaces

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.