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 [19,20]. 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.

中文翻译：

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临界Lebesgue空间中轴对称粘性Boussinesq系统的整体适定性

本文的贡献将集中在临界Lebesgue空间中轴对称粘性Boussinesq系统的三维情况下的整体存在性和唯一性主题。我们旨在推导最近在[]中获得的经典二维和三维轴对称Navier-Stokes方程的相似结果。19，20]。粗略地说，我们实质上表明，如果初始数据$（v_0，\ rho_0）$是轴对称的，并且$（\ omega_0，\ rho_0）$属于临界空间$ L ^ 1（\ Omega）\ times L ^ 1（ \ mathbb {R} ^ 3）$，其中$ \ omega_0 $是与$ v_0 $和$ \ Omega = \ {（r，z）\ in \ mathbb {R} ^ 2：r> 0 \相关的初始涡度} $，那么粘性的Boussinesq系统具有独特的全局解决方案。