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We present a formal derivation of a simplified version of CompressiblePrimitive Equations (CPEs) for atmosphere modeling. They are obtained from$3$-D compressible Navier-Stokes equations with an \emph{anisotropic viscousstress tensor} where viscosity depends on the density. We then study thestability of the weak solutions of this model by using an intermediate model,called model problem, which is more simple and practical, to achieve the mainresult.

Compressible primitive equation: formal derivation and stability of weak solutions