On the Derivation of the Compressible Primitive Equations

This chapter delves into the derivation of compressible primitive equations (CPE) from both Navier-Stokes and Euler equations, focusing on the mathematical intricacies and recent advancements. The text begins by introducing the primitive equation (PE) model in geophysical fluid dynamics, emphasizing its historical development and the challenges faced in its stability and accuracy. It then explores the rigorous derivation of CPE from compressible Navier-Stokes equations, utilizing relative entropy inequalities and hydrostatic approximations. The chapter also discusses the derivation of inviscid CPE from the Euler equations, highlighting the differences in mathematical structure and the unique challenges posed by compressible fluids. Additionally, the text provides a detailed analysis of the stability and existence of weak and strong solutions, offering insights into the global well-posedness of CPE in three-dimensional cases. The chapter concludes with a discussion on the implications of these derivations for long-term weather prediction and climate change studies, emphasizing the importance of understanding compressible fluid dynamics in atmospheric and oceanic modeling.