Well-posedness for the density-dependent incompressible flow of liquid crystals

In this paper, we are concerned with the Cauchy problem for the density-dependent incompressible flow of liquid crystals in thewhole space (N ≥ 2).We prove the localwell-posedness for large initial velocity field and director field of the system in critical Besov spaces if the initial density is close to a positive constant. We show also the global well-posedness for this system under a smallness assumption on initial data. In particular, this result allows us to work in Besov space with negative regularity indices, where the initial velocity becomes small in the presence of the strong oscillations.