Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity

This paper is concerned with the initial value problem of the three-dimensional nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity. We prove the existence of global strong solution on $\mathbb{R}^3 \times (0,\infty)$ under the initial norm ${\lVert u_0 \rVert}_{\dot{H}^\alpha} + {\lVert \nabla d_0 \rVert}_{\dot{H}^\alpha} (1/2 \lt \alpha \leq 1)$ being suitably small. In addition, the algebraic decay rate estimates of the global strong solution are obtained.