Abstract: In 1979, Zeng discussed the well-posedness problem of the atmospheric primitive equations without viscosity with Galerkin method in reference [209], and obtained the existence of weak solutions. In the early 1990s, many mathematicians (such as Lions, Temam and Wang) started the mathematical research of the primitive equations of large-scale atmosphere, oceans and the coupled atmospheric and oceanic primitive equations (see [140,141,142,143,144,203] and the references therein). In [140], through the introduction of viscosity and some technical processes, Lions, Temam and Wang obtained the new formulation of the large-scale dry atmospheric primitive equations suitable for mathematical treatments. In phase space H, the initial-boundary value problem of the new formulation of the primitive equations of the large-scale dry atmosphere is abbreviated as
where U = (v, T), for more detail, see section 3.1. In the pressure coordinate frame, this new formulation of the primitive equations resembles the Navier-Stokes equation of incompressible uid (of course there are some differences, such as, the nonlinear term of Navier-Stokes equation is (u · ∇)u, but the nonlinear term of the new primitive equations expression consists of
, where
u is the three-dimensional velocity field of Navier-Stokes equation, and
v is the horizontal velocity field of the atmosphere). By means of the methods used in studying Navier-Stokes equation in [137], they proved the global existence of weak solutions to the initial boundary value problem of the primitive equations (but they did not study the global well-posedness of the strong solutions). Under the assumption of global existence of strong solutions to the initial boundary value problem of the primitive equations of the atmosphere with vertical viscosity where the strong solutions satisfy the uniform boundedness of
H1 norm in time, they obtained Hausdorff and fractal dimension of the global attractors. With the same method, in references [141,142], they established the mathematical theories respectively about the primitive equations of the oceans and the coupled atmosphere-ocean model introduced in [142], where they mainly proved the global existence of weak solutions, and studied the estimates for the Hausdorff and fractal dimension estimations of the global attractors under the hypothesis of the global existence of strong solutions. In [130,131], in the hypothesis of global existence of strong solutions to the dry and moist atmospheric primitive equations, Li and Chou studied the asymptotic behavior of the corresponding solutions…
