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

Abstract: 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*}