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Abstract

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 a global regularity result for the 2D Boussinesq equations as long as $1-\alpha \lt \beta \lt \min\left\{\frac{\alpha}{2}, \frac{3\alpha-2}{2\alpha^{2}-6\alpha+5}, \frac{2-2\alpha}{4\alpha-3}\right\}$ with $0.77963\thickapprox\alpha_{0} \lt \alpha \lt 1$. This improves on some previous work.