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 R 3 × (0,∞) under the initial norm ∥u 0 ∥ Ḣα + ∥∇d 0 ∥ Ḣα (1/2 < α ≤ 1) being suitably small.In addition, the algebraic decay rate estimates of the global strong solution are obtained.