Remarks on global regularity of the 2D Boussinesq equations with fractional dissipation

In this paper, we are interested in the study of the Cauchy problem to the two-dimensional (2D) incompressible Boussinesq equations with fractional dissipation. By making use of the nonlinear lower bounds for the fractional Laplacian established in Constantin and Vicol (2012), we establish the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. This result significantly improves the recent works of Constantin and Vicol (2012) and Yang et al. (2014).