Global existence and exponential decay of strong solutions to the 3D nonhomogeneous nematic liquid crystal flows with density-dependent viscosity

In this paper, we consider an initial and boundary value problem to the three-dimensional (3D) nonhomogeneous nematic liquid crystal flows with density-dependent viscosity and vacuum. Combining delicate energy method with the structure of the system under consideration, the global well-posedness of strong solutions is established, provided that \(\Vert \rho _{0}\Vert _{L^{1}}+\Vert \nabla \varvec{d}_0\Vert _{L^2}\) $\Vert {\rho }_0{\Vert }_{L^1}+{\Vert \mathrm{\nabla }{\mathit{d}}_0\Vert }_{L^2}$ is suitably small. In particular, the initial velocity can be arbitrarily large. Moreover, the exponential decay rates of the strong solution are also obtained.