Derivation and stability study of a rigid lid bilayer model

Timack Ngom; Mamadou Sy

doi:10.3934/dcdsb.2011.16.361

Article Contents

2011, Volume 16, Issue 1: 361-383. Doi: 10.3934/dcdsb.2011.16.361

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Derivation and stability study of a rigid lid bilayer model

Timack Ngom^1, and
Mamadou Sy^2,

1.
Laboratoire d'Analyse Numérique et Informatique, Université Gaston Berger, UFR SAT BP 234 Saint-Louis, Sénégal, LAMA, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget du lac
2.
Laboratoire d'Analyse Numérique et d'Informatique (LANI), Université Gaston Berger, BP 234, Saint-Louis

Received: November 30, 2009

Revised: August 31, 2010

Published: June 2011

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Abstract

In this paper we present the derivation of a bilayer shallow water model with rigid lid hypothesis. We start from the incompressible Navier-Stokes equations, we introduce a small parameter $\varepsilon$ which is the ratio between the characteristic height and the characteristic length of the fluids domain. We use a formal asymptotic expansion then we resort to averaging to obtain the model. We also prove the stability of the model, in the following sense, up to a subsequence, every sequence of weak solutions converges to a solution of the model.

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Mathematics Subject Classification: Primary: 35Q35, 35Q30; Secondary: 35B35.

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