Global well-posedness to the 3D Cauchy problem of nonhomogeneous micropolar fluids involving density-dependent viscosity with large initial velocity and micro-rotational velocity

We show the global well-posedness to the three-dimensional (3D) Cauchy problem of nonhomogeneous micropolar fluids with density-dependent viscosity and vacuum in $\mathbb{R}^3$ provided that the initial mass is sufficiently small. Moreover, we also obtain that the gradients of velocity and micro-rotational velocity converge exponentially to zero in $H^1$ as time goes to infinity. Our analysis relies heavily on delicate energy estimates and the structural characteristic of the system under consideration. In particular, the initial velocity and micro-rotational velocity could be arbitrarily large.