Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system

Stefano Scrobogna

doi:10.3934/dcds.2017259

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2017, Volume 37, Issue 12: 5979-6034. Doi: 10.3934/dcds.2017259

This issue Previous Article Next Article Dynamics for a non-autonomous reaction diffusion model with the fractional diffusion

Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system

Stefano Scrobogna^1,2,

1.
BCAM -Basque Center for Applied Mathematics, Mazarredo, 14, E48009 Bilbao, Basque Country, Spain
2.
Institut de Mathématiques de Bordeaux, 351 Cours de la Libération, 33400 Talence, France

Received: February 2016

Revised: July 2017

Published: October 2017

This research was partially supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.

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Abstract

We consider a system describing the long-time dynamics of an hydrodynamical, density-dependent flow under the effects of gravitational forces. We prove that if the Froude number is sufficiently small such system is globally well posed with respect to a $ H^s, \ s>1/2 $ Sobolev regularity. Moreover if the Froude number converges to zero we prove that the solutions of the aforementioned system converge (strongly) to a stratified three-dimensional Navier-Stokes system. No smallness assumption is assumed on the initial data.

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Mathematics Subject Classification: Primary: 35Q30, 35Q35, 37N10, 76D03, 76D50, 76M45, 76M55.

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