Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Derivation and stability study of a rigid lid bilayer model

Pages: 361 - 383, Volume 16, Issue 1, July 2011      doi:10.3934/dcdsb.2011.16.361

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Timack Ngom - 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, France (email)
Mamadou Sy - Laboratoire d'Analyse Numérique et d'Informatique (LANI), Université Gaston Berger, BP 234, Saint-Louis, Senegal (email)

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.

Keywords:  Asymptotic expansion, rigid lid, stability, shallow water, bilayer.
Mathematics Subject Classification:  Primary: 35Q35, 35Q30; Secondary: 35B35.

Received: December 2009;      Revised: September 2010;      Available Online: April 2011.