From anisotropic Navier-Stokes equations to primitive equations for the ocean and atmosphere

TL;DR

This paper establishes the well-posedness of primitive equations for ocean and atmosphere modeling on specific domains and rigorously derives these from anisotropic Navier-Stokes equations, providing a solid mathematical foundation.

Contribution

It proves existence of global solutions for primitive equations and rigorously justifies their derivation from anisotropic Navier-Stokes equations in a unified functional setting.

Findings

Existence of global solutions in Besov spaces.

Rigorous derivation of primitive equations from anisotropic Navier-Stokes equations.

Validation of the limit process in the same functional framework.

Abstract

We study the well-posedness of the primitive equations for the ocean and atmosphere on two particular domains : a bounded domain $\Omega_1 := (-1, 1)^3$ with periodic boundary conditions and the strip $\Omega_2 := \mathbb{R}^2 \times (-1, 1)$ with a periodic boundary condition for the vertical coordinate. An existence theorem for global solutions on a suitable Besov space is derived. Then, in a second step, we rigorously justify the passage to the limit from the rescaled anisotropic Navier-Stokes equations to these primitive equations in the same functional framework as that found for the solutions of the primitive equations.