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*}
Categories: math.AP