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Global well-posedness and asymptotics for a penalized Boussinesq-type system without dispersion. (English) Zbl 1404.35447

Summary: 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 \(\mathrm F=1\) 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.

MSC:

35Q86 PDEs in connection with geophysics
35B45 A priori estimates in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76U05 General theory of rotating fluids
76B65 Rossby waves (MSC2010)
86A05 Hydrology, hydrography, oceanography
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