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 $$\|\frac{a_{0}}{r}\|_{\infty} \textrm{ and } \|u_{0}^θ\|_{3} \textrm{ are sufficiently small}. $$ Furthermore, if $\mathbf{u}_0\in L^1$ and $ru^θ_0\in L^1\cap L^2$, we have \begin{equation*} \|u^θ(t)\|_{2}^{2}+\langle t\rangle \|\nabla (u^θ\mathbf{e}_θ)(t)\|_{2}^{2}+t\langle t\rangle(\|u_{t}^θ(t)\|_{2}^{2}+\|Δ(u^θ\mathbf{e}_θ)(t)\|_{2}^{2}) \leq C \langle t\rangle^{-\frac{5}{2}},\ \forall\ t>0. \end{equation*}