Higher Order Regularity and Blow-up Criterion for Semi-dissipative and Ideal Boussinesq Equations

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.