The current paper is principally motivated by establishing the global well-posedness to the three-dimensional Boussinesq system with zero diffusivity in the setting of axisymmetric flows without swirling with $v_0\in H^{\frac{1}{2}}\left({\mathbb{R}}^3\right)\cap {\dot{B}}_{3,1}^0\left({\mathbb{R}}^3\right)$ and density ${\rho }_0\in L^2\left({\mathbb{R}}^3\right)\cap {\dot{B}}_{3,1}^0\left({\mathbb{R}}^3\right)$. This respectively enhances the two results recently accomplished in Danchin and Paicu (2008) and Hmidi and Rousset (2010). Our formalism is inspired, in particular for the first part from Abidi (2008) concerning the axisymmetric Navier–Stokes equations once $v_0\in H^{\frac{1}{2}}\left({\mathbb{R}}^3\right)$ and external force $f\in L_{loc}^2({\mathbb{R}}_{+};H^{\beta }\left({\mathbb{R}}^3\right))$, with $\beta >\frac{1}{4}$. This latter regularity on $f$ which is the density in our context is helpless to achieve the global estimates for Boussinesq system. This technical defect forces us to deal once again with a similar proof to that of Abidi (2008) but with $f\in L_{loc}^{\beta }({\mathbb{R}}_{+};L^2\left({\mathbb{R}}^3\right))$ for some $\beta >4$. Second, we explore the gained regularity on the density by considering it as an external force in order to apply the study already obtained to the Boussinesq system.

本文的主要动机是通过在轴对称流动的情况下建立零扩散的零扩散率的三维Boussinesq系统的整体适定性来实现的。 $v_0\in H^{\frac{\mathrm{1个}}{2}}（{\mathbb{[R}}^3）\cap {\dot{乙}}_{3，\mathrm{1个}}^0（{\mathbb{[R}}^3）$ 和密度 ${\rho }_0\in {\mathrm{大号}}^2（{\mathbb{[R}}^3）\cap {\dot{乙}}_{3，\mathrm{1个}}^0（{\mathbb{[R}}^3）$。这分别增强了Danchin和Paicu（2008）以及Hmidi和Rousset（2010）最近完成的两个结果。我们的形式主义受到启发，特别是在Abidi（2008）的第一部分中，曾经涉及轴对称的Navier-Stokes方程。$v_0\in H^{\frac{\mathrm{1个}}{2}}（{\mathbb{[R}}^3）$ 和外力 $F\in {\mathrm{大号}}_{升ØC}^2（{\mathbb{[R}}_{+};H^{\beta }（{\mathbb{[R}}^3））$，带有 $\beta >\frac{\mathrm{1个}}{4}$。后者的规律性$F$在我们的上下文中，这是密度，无法实现Boussinesq系统的全局估计。这种技术缺陷迫使我们再次处理与Abidi（2008）类似的证明，但$F\in {\mathrm{大号}}_{升ØC}^{\beta }（{\mathbb{[R}}_{+};{\mathrm{大号}}^2（{\mathbb{[R}}^3））$ 对于一些 $\beta >4$。其次，为了将已经获得的研究应用到Boussinesq系统中，我们将其视为外力来探索密度的规律性。