Highly rotating fluids with vertical stratification for periodic data and vanishing vertical viscosity

TL;DR

This paper proves the global well-posedness and convergence of solutions for primitive equations with vertical stratification and no vertical diffusivity, under small Rossby and Froude numbers, in anisotropic Sobolev spaces.

Contribution

It establishes the global well-posedness and convergence results for primitive equations with specific anisotropic conditions and domain types, extending previous understanding of these fluid models.

Findings

Global well-posedness of primitive equations without vertical diffusivity.

Convergence of solutions as Rossby and Froude numbers tend to zero.

Applicability to a broad class of tori including non-resonant and many resonant tori.

Abstract

We prove that the primitive equations without vertical diffusivity are globally well-posed (if the Rossby and Froude number are sufficiently small) in suitable Sobolev anisotropic spaces. Moreover if the Rossby and Froude number tend to zero at a comparable rate the global solutions of the primitive equations converge globally to the global solutions of a suitable limit system. The space domain considered has to belong to a class of tori which is general enough to include all non-resonant tori and many resonant tori as well.