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On the Cauchy Problem of 3D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

We consider the global existence and large-time asymptotic behavior of strong solutions to the Cauchy problem of the three-dimensional nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity and vacuum. We establish some key a priori exponential decay-in-time rates of the strong solutions. Then after using these estimates, we also obtain the global existence of strong solutions in the whole three-dimensional space, provided that the initial velocity is suitably small in the $\dot H^β$-norm for some $β\in(1/2,1].$ Note that this result is proved without any smallness conditions on the initial density. Moreover, the density can contain vacuum states and even have compact support initially.