  The contribution of this paper will be focused on the global existence and
uniqueness topic in three-dimensional case of the axisymmetric viscous
Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous
results for the classical two-dimensional and three-dimensional axisymmetric
Navier-Stokes equations recently obtained in \cite{Gallay,Gallay-Sverak}.
Roughly speaking, we show essentially that if the initial data $(v_0,\rho_0)$
is axisymmetric and $(\omega_0,\rho_0)$ belongs to the critical space
$L^1(\Omega)\times L^1(\mathbb{R}^3)$, with $\omega_0$ is the initial vorticity
associated to $v_0$ and $\Omega=\{(r,z)\in\mathbb{R}^2:r>0\}$, then the viscous
Boussinesq system has a unique global solution.
