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

J.-Y. Chemin proved the convergence (as the Rossby number $\varepsilon$ 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 $\nu^{\prime}$ 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.

Keywords

geophysical fluids, primitive equations, Boussinesq system, 3D-quasi-geostrophic system

2010 Mathematics Subject Classification

35A01, 35B45, 35Q86, 76D03, 76U05

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This work was supported by the ANR project INFAMIE, ANR-15-CE40-0011.

Received 23 July 2017

Received revised 17 February 2018

Accepted 17 February 2018

Published 30 August 2018