This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by Λ α u in the velocity equation and by Λ β θ in the temperature equation, where Λ−−Δ denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when (α, β) is in the critical range: α>(1777−23)/24=0.789103.., β > 0, and α + β = 1. This result improves previous work which obtained the global regularity for α>(23−145)/12≈0.9132,β>0, and α + β = 1.
