# American Institute of Mathematical Sciences

December  2017, 37(12): 5979-6034. doi: 10.3934/dcds.2017259

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

 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  August 2017

Fund Project: 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

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.

Citation: Stefano Scrobogna. Derivation of limit equations for a singular perturbation of a 3D periodic Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 5979-6034. doi: 10.3934/dcds.2017259
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