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) x H-1 improves considerably the main result of the recent article (Hu et al. in J Math Phys 54(8): 081507, 2013). Our second result on global boundedness of the solution (u, theta) in V x H-1 for both bounded domain and the whole space 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) x H-2 is also proved.