Justification of the Hydrostatic Approximation of the Primitive Equations in Anisotropic Space $L^\infty_H L^q_{x_3}(\Torus^3)$

TL;DR

This paper rigorously justifies the hydrostatic approximation of the primitive equations derived from scaled Navier-Stokes equations within anisotropic function spaces, enhancing understanding of their validity in geophysical fluid models.

Contribution

It provides a mathematical justification for the hydrostatic approximation in anisotropic spaces, a key step in validating primitive equations for geophysical applications.

Findings

Hydrostatic approximation holds in $L^ abla_H L^q_{x_3}$ spaces for $q \\geq 1$.

The results extend the validity of primitive equations in anisotropic functional frameworks.

The paper bridges the gap between Navier-Stokes and primitive equations in specific anisotropic spaces.

Abstract

The primitive equations are fundamental models in geophysical fluid dynamics and derived from the scaled Navier-Stokes equations. In the primitive equations, the evolution equation to the vertical velocity is replaced by the so-called hydrostatic approximation. In this paper, we give a justification of the hydrostatic approximation by the scaled Navier-Stoke equations in anisotropic spaces $L^\infty_H L^q_{x_3} (\Torus^3)$ for $q \geq 1$.