A note on global regularity results for 2D Boussinesq equations with fractional dissipation

In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the global regularity result of the 2D Boussinesq equations as long as $1-\alpha<\beta< \min\Big\{\frac{\alpha}{2},\,\, \frac{3\alpha-2}{2\alpha^{2}-6\alpha+5}, \,\,\frac{2-2\alpha}{4\alpha-3}\Big\}$ with $0.77963\thickapprox\alpha_{0}<\alpha<1$. As a result, this result is a further improvement of the previous two works \cite{MX,YXX}.