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 Hs for s>n2 to the viscous, zero thermal-diffusive Boussinesq equations in Rn,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 for nonviscous and fully ideal Boussinesq systems in Rn,n=2,3. Moreover, we establish blow-up criterion for nonviscous Boussinesq system in three dimensions and for fully ideal Boussinesq system in both two and three dimensions. Commutator estimates from the work of Kato and Ponce [Comm. Pure Appl. Math. 41, 891 (1988)] and Fefferman et al. [J. Funct. Anal. 267, 1035 (2014)] play important roles in the calculations.