On the Global Existence for the Axisymmetric Euler-Boussinesq System in Critical Besov Spaces

This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data $v^0\in B_{2,1}^{5\mathrm{/}2}({\text{RR}}^3)$ and$ ${\rho}^{0}{\in}B_{2,1}^{1/2}(\RR^3)\cap L^{p}(\RR^3)$with$p>6.$ This system couples the incompressible Euler equations with a transport-diffusion equation governing the density. In this case the Beale-Kato-Majda criterion is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity.