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Justification of the PE diagram and singular limit

I will introduce the PE diagram, which describe the asymptotic relations between compressible hydrodynamic equations, incompressible hydrodynamic equations, primitive equations, and compressible primitive equations. We will review current developments of the ocean scale, and introduce our contribution to the atmosphere scale in the PE diagram. In particular, our works of the well-posedness and asymptotic limits of the compressible primitive equations will fill the gaps in the PE diagram. We will detour to singular limit to illustrate the goal and tool in our study.