On the global well-posedness of axisymmetric viscous Boussinesq system in critical Lebesgue spaces

Abstract

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.