Abstract
<p style='text-indent:20px;'>The incompressible Boussinesq system plays an important role in modelling geophysical fluids and studying the Raleigh-Bernard convection. We consider the regularized model (also named as Boussinesq-<inline-formula><tex-math id="M1">$ \alpha $</tex-math></inline-formula> model) to the Boussinesq equations. We consider the Cauchy problem of a two-dimensional regularized Boussinesq model with vertical dissipation in the horizontal regularized velocity equation and horizontal dissipation in the vertical regularized velocity equation and prove that this system has a unique global classical solution. Next, we consider a two-dimensional Boussinesq-<inline-formula><tex-math id="M2">$ \alpha $</tex-math></inline-formula> model with only vertical thermal diffusion and establish a Beale-Kato-Majda type regularity condition of smooth solution for this system.</p>
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