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| In this talk, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong solutions are established for this system with $H^2$ initial data. Furthermore, we will show that for certain class of initial data the 2D and 3D primitive equations are either ill-posed or develop finite-time shock-like singularities. |
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