In this talk I will show the global (in time) well-posedness for the 3D viscous primitive equations of atmospheric and oceanic dynamics for all initial data. Motivated by strong anisotropic turbulence mixing I will also show the global well-posedness of this model with only horizontal viscosity and diffusion. Similar results also hold for the case of full viscosity, but only vertical diffusion. On the other hand, I will show that in the inviscid case there is a class of initial data for which the corresponding smooth solutions of the inviscid primitive equations develop singulary (blow-up) in finite-time.

 This talk is based on different joint results with C. Cao, S. Ibrahim, J. Li and K. Nakanishi.