Global solvability of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity

Abstract: In this paper, we consider 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 $|\nabla u_0|_{L^2}$ is suitably small with arbitrary large initial density. This generalizes all the previous results even for the constant viscosity.