Global solutions of 3D axisymmetric Boussinesq equations with nonzero swirl

Considering the 3D axisymmetric Boussinesq system with nonzero swirl, we obtain the global existence and uniqueness of the strong solutions (u,ρ), when ‖rdu0θ‖L31−d, d∈[0,1], is sufficiently small. Furthermore, if u0∈L32(R3) and ru0θ∈L1(R3)∩L2(R3), we have the decay estimate ‖u(t)‖L2(R3)2+〈t〉2‖ρ(t)‖L2(R3)2+〈t〉2‖uθ(t)‖L2(R3)2≤C〈t〉−12,for any t>0. At last, we get several continuation criteria.