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The nonhomogeneous Navier-Stokes equations with density-dependent viscosityis studied in three-dimensional (3D) exterior domains with nonslip or slipboundary conditions. We prove that the strong solutions exists globally in timeprovided that the gradient of the initial velocity is suitably small. Here theinitial density is allowed to contain vacuum states. Moreover, after developingsome new techniques and methods, the large-time behavior of the strongsolutions with exponential decay-in-time rates is also obtained.

Global Strong Solutions to Density-Dependent Viscosity Navier-Stokes Equations in 3D Exterior Domains