On the hydrostatic approximation of compressible anisotropic Navier-Stokes equations -- rigorous justification

TL;DR

This paper rigorously justifies the hydrostatic approximation of compressible Navier-Stokes equations in large-scale geophysical flows by employing a relative entropy inequality to derive the compressible Primitive Equations.

Contribution

It is the first work to use the relative entropy inequality for proving hydrostatic approximation and deriving the compressible Primitive Equations.

Findings

Established the limit from compressible Navier-Stokes to Primitive Equations.

Applied relative entropy inequality in a novel way for hydrostatic approximation.

Provided rigorous mathematical justification for the hydrostatic limit in compressible flows.

Abstract

In this work, we obtain the hydrostatic approximation by taking the small aspect ratio limit to the Navier-Stokes equations. 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. This is the first work to use relative entropy inequality for proving hydrostatic approximation and derive the compressible Primitive Equations.