tailieunhanh - Đề tài " Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics "

In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics. 1. Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations, with rotation, coupled to thermodynamics and salinity diffusion-transport equations, which account for the buoyancy forces and stratification effects under the Boussinesq approximation. . | Annals of Mathematics Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics By Chongsheng Cao and Edriss S. Titi Annals of Mathematics 166 2007 245 267 Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics By Chongsheng Cao and Edriss S. Titi Abstract In this paper we prove the global existence and uniqueness regularity of strong solutions to the three-dimensional viscous primitive equations which model large scale ocean and atmosphere dynamics. 1. Introduction Large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the Navier-Stokes equations with rotation coupled to thermodynamics and salinity diffusion-transport equations which account for the buoyancy forces and stratification effects under the Boussinesq approximation. Moreover and due to the shallowness of the oceans and the atmosphere . the depth of the fluid layer is very small in comparison to the radius of the earth the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one which in turn leads to modeling the vertical motion by the hydrostatic balance. As a result one obtains the system 1 4 which is known as the primitive equations for ocean and atmosphere dynamics see . 20 21 22 23 24 33 and references therein . We observe that in the case of ocean dynamics one has to add the diffusion-transport equation of the salinity to the system 1 - 4 . We omitted it here in order to simplify our mathematical presentation. However we emphasize that our results are equally valid when the salinity effects are taking into account. Note that the horizontal motion can be further approximated by the geostrophic balance when the Rossby number the ratio of the horizontal acceleration to the Coriolis force is very small. By taking advantage of these assumptions and other geophysical .