Global strong solution of 3D inhomogeneous Navier–Stokes equations with density-dependent viscosity

In this paper, we investigate the three-dimensional inhomogeneous Navier–Stokes equations with density-dependent viscosity in presence of vacuum over bounded domains. Global-in-time unique strong solution is proved to exist when ‖∇u0‖L2‖∇u0‖L2 is suitably small with arbitrary large initial density.