Global axisymmetric solutions of 3D inhomogeneous incompressible Navier-Stokes Systems with nonzero swirl

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 $$\mathrm{\parallel }\frac{a_0}{r}{\mathrm{\parallel }}_{\mathrm{\infty }}\text{ and }\mathrm{\parallel }{u}_0^{\theta }{\mathrm{\parallel }}_3\text{ are sufficiently small}\mathrm{.}$$ Furthermore, if ${\mathbf{u}}_0\in L^1$ and $r{u}_0^{\theta }\in L^1\cap L^2$, 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*}