Global strong solution to the two-dimensional density-dependent nematic liquid crystal flows with vacuum

We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy $\|\sqrt{\rho_0}\mathbf{u}_0\|_{L^2}^2+\|\nabla\mathbf{d}_0\|_{L^2}^2$ is small. In particular, the initial density may contain vacuum states and even have compact support. Moreover, the large time behavior of the solution is also obtained.
Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:1507.06471, arXiv:1506.03884, Nonlinearity, 2017