  In this paper we establish local-in-time existence and uniqueness of strong
solutions in $H^s$ for $s > \frac{n}{2}$ to the viscous, zero thermal-diffusive
Boussinesq equations in $\mathbb{R}^n , n = 2,3$. Beale-Kato-Majda type blow-up
criterion has been established in three-dimensions with respect to the
$BMO$-norm of the vorticity. We further prove the local-in-time existence and
blow-up criterion for non-viscous and fully ideal Boussinesq systems.
Commutator estimates due to Kato and Ponce (1988) \cite {KP} and Fefferman et.
al. (2014) \cite {Fe} play important roles in the calculations.
