We study the global regularity of solutions to the 2D
Boussinesq equations with fractional dissipation, given by
${\left(\Delta \right)}^{\alpha \u22152}u$ in the velocity
equation and by
${\left(\Delta \right)}^{\beta \u22152}\theta $
in the temperature equation. We establish the global regularity for
$\frac{2}{3}<\alpha <1$,
$\alpha +\beta >1$ and
$\alpha >\frac{1}{1+\beta}$. This result is for the
subcritical regime
$\alpha +\beta >1$
and the point here is to obtain the global regularity for the largest possible range of
$\alpha $.
