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In this paper, we consider the three-dimensional inhomogeneous Navier-Stokesequations with density-dependent viscosity in presence of vacuum over boundeddomains. 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.

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