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

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) is an element of H-1/2 (R-3) boolean AND (B) over dot(3,1)(0) (R-3) and density rho(0) is an element of L-2(R-3) boolean AND (B) over dot(3,1)(0)(R-3). 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) is an element of H-1/2 (R-3) and external force f is an element of L-loc(2) (R+; H-beta(R-3)), with beta > 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 is an element of L-loc(beta) (R+; L-2(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. (C) 2020 Elsevier Ltd. All rights reserved.