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July  2011, 16(1): 361-383. doi: 10.3934/dcdsb.2011.16.361

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, France
2. 

Laboratoire d'Analyse Numérique et d'Informatique (LANI), Université Gaston Berger, BP 234, Saint-Louis

Received  December 2009 Revised  September 2010 Published  April 2011

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: bilayer., rigid lid, stability, shallow water, Asymptotic expansion.
Mathematics Subject Classification: Primary: 35Q35, 35Q30; Secondary: 35B3.
Citation: Timack Ngom, Mamadou Sy. Derivation and stability study of a rigid lid bilayer model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 361-383. doi: 10.3934/dcdsb.2011.16.361
References:
[1]

F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography,, Commun. Math. Sci., 2 (2004), 359.   Google Scholar

[2]

E. Audusse, A multilayer Saint-Venant model: derivation and numerical validation,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 189.  doi: 10.3934/dcdsb.2005.5.189.  Google Scholar

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B. Di Martino, P. Orenga and M. Peybernes, On a bi-layer shallow water model with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 15 (2005), 843.  doi: 10.1142/S0218202505000583.  Google Scholar

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G. Narbona-Reina, J. D. D. Zabsonré, E. D. Fernández-Nieto and D. Bresch, Derivation of a bilayer model for shallow water equations with viscosity. Numerical validation,, CMES Comput. Model. Eng. Sci., 43 (2009), 27.   Google Scholar

[6]

B. Di Martino, C. Giacomoni and P. Orenga, Analysis of some shallow water problems with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 11 (2001), 979.  doi: 10.1142/S0218202501001203.  Google Scholar

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Philippe Guyenne, David Lannes and Jean-Claude Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves,, Nonlinearity, 23 (2010), 237.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

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D. Bresch and M. Renardy, Well-posedness of two-layer shallow water flow between two horizontal rigid plates,, To appear in nonlinearity, (2011).   Google Scholar

[10]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,, Comm. Math. Phys., 238 (2003), 211.   Google Scholar

[11]

J. Simon, "Équation de Navier-Stokes,'', Cours de DEA 2002-2003 Universit茅 Blaise Pascal Clermont-Ferrand., (): 2002.   Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,'', Dunod, (1969).   Google Scholar

[13]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations,, Comm. Partial Differential Equations, 32 (2007), 431.  doi: 10.1080/03605300600857079.  Google Scholar

[14]

D. Bresch, B. Desjardins and D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains,, J. Math. Pures Appl. (9), 87 (2007), 227.  doi: 10.1016/j.matpur.2006.10.010.  Google Scholar

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F. Boyer and P. Fabrie, "Eléments D'analyse pour L'éetude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles,'', Mathématiques & Applications (Berlin) [Mathematics & Applications]., ().   Google Scholar

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F. Boyer, "Analyse Numérique des edp Elliptiques,'', Cours Master 2 2009 Université Paul Cézanne., (2009).   Google Scholar

show all references

References:
[1]

F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography,, Commun. Math. Sci., 2 (2004), 359.   Google Scholar

[2]

E. Audusse, A multilayer Saint-Venant model: derivation and numerical validation,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 189.  doi: 10.3934/dcdsb.2005.5.189.  Google Scholar

[3]

E. Bruce Pitman and Long Le, A two-fluid model for avalanche and debris flows,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 363 (2005), 1573.   Google Scholar

[4]

B. Di Martino, P. Orenga and M. Peybernes, On a bi-layer shallow water model with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 15 (2005), 843.  doi: 10.1142/S0218202505000583.  Google Scholar

[5]

G. Narbona-Reina, J. D. D. Zabsonré, E. D. Fernández-Nieto and D. Bresch, Derivation of a bilayer model for shallow water equations with viscosity. Numerical validation,, CMES Comput. Model. Eng. Sci., 43 (2009), 27.   Google Scholar

[6]

B. Di Martino, C. Giacomoni and P. Orenga, Analysis of some shallow water problems with rigid-lid hypothesis,, Math. Models Methods Appl. Sci., 11 (2001), 979.  doi: 10.1142/S0218202501001203.  Google Scholar

[7]

María Luz Muñoz-Ruiz, Manuel Jesú Castro-Díaz and Carlos Parés, On an one-dimensional bi-layer shallow-water problem,, Nonlinear Anal., 53 (2003), 567.  doi: 10.1016/S0362-546X(02)00137-2.  Google Scholar

[8]

Philippe Guyenne, David Lannes and Jean-Claude Saut, Well-posedness of the Cauchy problem for models of large amplitude internal waves,, Nonlinearity, 23 (2010), 237.  doi: 10.1088/0951-7715/23/2/003.  Google Scholar

[9]

D. Bresch and M. Renardy, Well-posedness of two-layer shallow water flow between two horizontal rigid plates,, To appear in nonlinearity, (2011).   Google Scholar

[10]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model,, Comm. Math. Phys., 238 (2003), 211.   Google Scholar

[11]

J. Simon, "Équation de Navier-Stokes,'', Cours de DEA 2002-2003 Universit茅 Blaise Pascal Clermont-Ferrand., (): 2002.   Google Scholar

[12]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,'', Dunod, (1969).   Google Scholar

[13]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations,, Comm. Partial Differential Equations, 32 (2007), 431.  doi: 10.1080/03605300600857079.  Google Scholar

[14]

D. Bresch, B. Desjardins and D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains,, J. Math. Pures Appl. (9), 87 (2007), 227.  doi: 10.1016/j.matpur.2006.10.010.  Google Scholar

[15]

F. Boyer and P. Fabrie, "Eléments D'analyse pour L'éetude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles,'', Mathématiques & Applications (Berlin) [Mathematics & Applications]., ().   Google Scholar

[16]

F. Boyer, "Analyse Numérique des edp Elliptiques,'', Cours Master 2 2009 Université Paul Cézanne., (2009).   Google Scholar

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