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

New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution $(u, \theta) \in D(A)\times H^1$ 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 a new one. It has been open and also seems much more challenging than the first result. Global regularity of the solution $(u, \theta) \in D(A) \times H^2$ is also proved.