We consider the Cauchy problems for a two-dimensional nonlinear system of differential equations arising from geostrophics. Recently, we find that there exist a few special structures in the system. By making complete use of the special structures, we establish the uniform energy estimates for all order derivatives to accomplish the existence of the global smooth solution. Moreover, by using the Fourier splitting method, we accomplish the decay estimates with sharp rates. Furthermore, by using  many traditional ideas and the decay estimates, we establish the exact limits of the global smooth solution.

海报