Global smooth solution to the 2D Boussinesq equations with fractional dissipation

In this paper, we consider the two-dimensional (2D) incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. Based on the previous works and some new observations, we show that the condition $1-α<β<\min\Big\{3-3α,\,\,\fracα{2},\,\, \frac{3α^{2}+4α-4}{8(1-α)}\Big\}$ with $0.7351\approx\frac{10-2\sqrt{10}}{5}<α<1$ suffices in order for the solution pair of velocity and temperature to remain smooth for all time.

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