Rigorous justification of hydrostatic approximation of compressible fluid flow equations

In this paper, we obtain the 3D compressible primitive equations approximation with gravity by taking the small aspect ratio limit to the Navier-Stokes equations. We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier-Stokes equations with a pressure law of the form p(ρ) = ρ2 to the compressible primitive equations.