Global strong solution of the 3D inhomogeneous liquid crystal flows with density-dependent viscosity and large velocity

This paper concerns the initial boundary value problem of three-dimensional inhomogeneous incompressible liquid crystal flows with density-dependent viscosity. When the viscosity coefficient $\mu (\rho )$ is a power function of the density with the power larger than $1$, that is $\mu (\rho )=\mu {\rho }^{\alpha }$ with $\alpha >1$, it is proved that the system exists a unique global strong solution as long as the initial density is sufficiently large and $L^3$-norm of the derivative of the initial director is sufficiently small. This is the first result concerning the global strong solution for three-dimensional inhomogeneous liquid crystal flows without smallness of velocity.