  In this paper, we investigate the global well-posedness for the 3-D
inhomogeneous incompressible Navier-Stokes system with the axisymmetric initial
data. We prove the global well-posedness provided that
  $$\|\frac{a_{0}}{r}\|_{\infty} \textrm{ and } \|u_{0}^{\theta}\|_{3} \textrm{
are sufficiently small}.
  $$
  Furthermore, if $\mathbf{u}_0\in L^1$ and $ru^\theta_0\in L^1\cap L^2$, we
have \begin{equation*} \|u^{\theta}(t)\|_{2}^{2}+\langle t\rangle \|\nabla
(u^{\theta}\mathbf{e}_{\theta})(t)\|_{2}^{2}+t\langle
t\rangle(\|u_{t}^{\theta}(t)\|_{2}^{2}+\|\Delta(u^{\theta}\mathbf{e}_{\theta})(t)\|_{2}^{2})
\leq C \langle t\rangle^{-\frac{5}{2}},\ \forall\ t>0. \end{equation*}
