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

Abstract

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 $v_0\in H^{\frac{1}{2}}({\mathbb{R}}^3)\cap {\dot{B}}_{3,1}^0({\mathbb{R}}^3)$ and density ${\rho }_0\in L^2({\mathbb{R}}^3)\cap {\dot{B}}_{3,1}^0({\mathbb{R}}^3)$. This respectively enhances the two results recently accomplished in \cite{Danchin-Paicu1, Hmidi-Rousset}. Our formalism is inspired, in particular for the first part from \cite{Abidi} concerning the axisymmetric Navier-Stokes equations once $v_0\in H^{\frac{1}{2}}({\mathbb{R}}^3)$ and external force $f\in L_{loc}^2({\mathbb{R}}_{+};H^{\beta }({\mathbb{R}}^3))$, 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 \cite{Abidi} but with $f\in L_{loc}^{\beta }({\mathbb{R}}_{+};L^2({\mathbb{R}}^3))$ 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.