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New results are obtained for global regularity and long-time behavior of thesolutions to the 2D Boussinesq equations for the flow of an incompressiblefluid with positive viscosity and zero diffusivity in a smooth bounded domain.Our first result for global boundedness of the solution $(u, \theta) \inD(A)\times H^1$ improves considerably the main result of the recent article[7]. Our second result on global regularity of the solution $(u, \theta) \in V\times H^1$ for both bounded domain and the whole space ${\mathbb R}^2$ is anew one. It has been open and also seems much more challenging than the firstresult. Global regularity of the solution $(u, \theta) \in D(A) \times H^2$ isalso proved.

Global Regularity and Long-time Behavior of the Solutions to the 2D Boussinesq Equations without Diffusivity in a Bounded Domain