Global well-posedness and asymptotics for a penalized Boussinesq-type system without dispersion

TL;DR

This paper establishes global well-posedness and asymptotic behavior for a penalized Boussinesq-type system without dispersion, extending previous convergence results of primitive equations to the quasi-geostrophic system for general viscosities.

Contribution

It generalizes Chemin's convergence result to any viscosities by leveraging the quasi-geostrophic structure, removing the need for dispersive properties.

Findings

Proves global well-posedness of the penalized Boussinesq system.

Shows convergence to the quasi-geostrophic system as Rossby number tends to zero.

Extends previous results to a broader viscosity regime.

Abstract

J.-Y. Chemin proved the convergence (as the Rossby number $\epsilon$ goes to zero) of the solutions of the Primitive Equations to the solution of the 3D quasi-geostrophic system when the Froude number F = 1 that is when no dispersive property is available. The result was proved in the particular case where the kinematic viscosity $\nu$ and the thermal diffusivity $\nu$ ' are close. In this article we generalize this result for any choice of the viscosities, the key idea is to rely on a special feature of the quasi-geostrophic structure.