Global solution to the 3-D density-dependent incompressible flow of liquid crystals

In this paper, we consider the global wellposedness of the 3-D density-dependent incompressible flow of liquid crystals in the critical Besov spaces. We first get the local solution for this system without assumptions of small density variation and then extend this local solution to be global under the smallness assumptions on initial density, initial orientation field and the horizontal components of the initial velocity field. This result implies the global well-posedness of this system with large initial vertical velocity component.