Persistence of global well-posedness for the 2D Boussinesq equations with fractional dissipation

In this paper, we study the IBVP for the 2D Boussinesq equations with fractional dissipation in the subcritical case, and prove the persistence of global well-posedness of strong solutions. Moreover, we also prove the long time decay of the solutions, and investigate the existence of the solutions in Sobolev spaces $W^{2,p}({R}^{2})\times W^{1,p}({R}^{2})$ W 2 , p ( R 2 ) × W 1 , p ( R 2 ) for some $p>2$ p > 2 .