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The current paper is principally motivated by establishing the globalwell-posedness to the three-dimensional Boussinesq system with zero diffusivityin the setting of axisymmetric flows without swirling with $v_0\inH^{\frac12}(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$ and density$\rho_0\in L^2(\mathbb{R}^3)\cap \dot{B}^{0}_{3,1}(\mathbb{R}^3)$. Thisrespectively enhances the two results recently accomplished in\cite{Danchin-Paicu1, Hmidi-Rousset}. Our formalism is inspired, in particularfor the first part from \cite{Abidi} concerning the axisymmetric Navier-Stokesequations once $v_0\in H^{\frac12}(\mathbb{R}^3)$ and external force $f\inL^2_{loc}\big(\mathbb{R}_{+};H^{\beta}(\mathbb{R}^3)\big)$, with$\beta>\frac14$. This latter regularity on $f$ which is the density in ourcontext is helpless to achieve the global estimates for Boussinesq system. Thistechnical defect forces us to deal once again with a similar proof to that of\cite{Abidi} but with $f\inL^{\beta}_{loc}\big(\mathbb{R}_{+};L^2(\mathbb{R}^3))$ for some $\beta>4$.Second, we explore the gained regularity on the density by considering it as anexternal force in order to apply the study already obtained to the Boussinesqsystem.

On the global solvability of the axisymmetric Boussinesq system with critical regularity