In this paper, we obtain the 3D compressible primitive equations approximation without gravity by taking the small aspect ratio limit to the Navier-Stokes equations in the isothermal case with gravity. The aspect ratio (the ratio of the depth to horizontal width) is a geometrical constraint in general large scale geophysical motions that the vertical scale is significantly smaller than horizontal. We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier-Stokes equations to the compressible primitive equations. In addition to the presence of gravity, we consider that the viscosity of the fluid depends on its density and that it is submitted to a quadratic friction force.

anisotropic Navier-Stokes equations, aspect ratio limit, compressible primitive equations

Received: November 19, 2023;  Accepted: January 25, 2024; Published: February 28, 2024

How to cite this article: Jules Ouya and Arouna Ouedraogo, On 3D compressible primitive equations approximation of anisotropic Navier-Stokes equations: rigorous justification, Advances in Differential Equations and Control Processes 31(1) (2024), 109-151. http://dx.doi.org/10.17654/0974324324007

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