Global Fujita-Kato solutions to 3D inhomogeneous nematic liquid crystal flows with only bounded density allowing vacuum

This paper is devoted to proving the global existence and uniqueness of weak solutions to the three dimensional inhomogeneous incompressible nematic liquid crystal flows provided that the initial density is in the bounded function space with vacuum and the initial velocity is sufficiently small in the critical Sobolev space $ \dot{H}^{\frac12}(\mathbb{R}^3) $ and the gradient of initial molecule orientation is also small enough in the critical Sobolev space $ \dot{H}^{\frac12}(\mathbb{R}^3) $. Our results improves the existence and uniqueness result established by Francesco. It also corresponds to the Fujita-Kato solution to the inhomogeneous incompressible Navier-Stokes equation which was established by Hao, Shao, Wei and Zhang.