On the Cauchy problem of 3D nonhomogeneous micropolar fluids with density-dependent viscosity

In this paper, we considered the global well-posedness of strong solutions to the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible micropolar fluids with density-dependent viscosity and vacuum. Based on the energy method, some key a priori exponential decay-in-time rates of strong solutions are obtained. As a result, the existence and large-time asymptotic behavior of strong solutions in the whole space ${\mathbb{R}}^3$ are established, provided that the initial mass is sufficiently small. Note that this result is proven without any compatibility conditions.