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October  2016, 9(5): 1521-1551. doi: 10.3934/dcdss.2016061

Coupling the shallow water equation with a long term dynamics of sand dunes

Mouhamadou Aliou M. T. Baldé 1, and  Diaraf Seck 1,

1. 

Université Cheikh Anta Diop de Dakar, BP 16889 Dakar Fann , Ecole Doctorale de Mathématiques et Informatique, Laboratoire de Mathématiques de la Décision et d'Analyse Numérique, (L.M.D.A.N) F.A.S.E.G, Senegal

Received  February 2015 Revised  April 2016 Published  October 2016

In this paper we couple a long term dynamic equation of dunes of sand(LTDD) in [6] with a shallow water equation(SWE). And we study the evolution of sand dunes over long periods in the marine environment near the coast. We use works due to S. Klainerman & A. Majda [9] to show on the one hand existence and uniqueness results. On the other hand we give estimations of solutions for the dimensionless coupled system SWE-LTDD. And finally the coupled system is homogenized.
Keywords: homogenization., Shallow water equations, asymptotics, singular PDE, hyperbolic, parabolic, sand dunes.
Mathematics Subject Classification: 35L45, 58J45, 35K15, 58J35, 58J3.
Citation: Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061
References:
[1]

G. Allaire, Homogenization and Two-Scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.

[2]

P. Alliot, E. Frénod and V. Monbet, Modelling the coastal ocean over a time period of several weeks, Journal of Differential Equations, 248 (2010), 639-659. doi: 10.1016/j.jde.2009.11.004.

[3]

E. Audusse, F. Benkhadloun, J. Sainte-Marie and M. Seaid, Multilayer Saint-Venant equations over movable beds, Discrete and Continuous Dynamical Systems - B, 15 (2011), 917-934. doi: 10.3934/dcdsb.2011.15.917.

[4]

C. Berthon, S. Cordier, O. Delestre and M. H. Le, An analytical solution of shallow water system coupled to Exner equation, Comptes Rendus Mathématique, Elsevier, 350 (2012), 183-186. doi: 10.1016/j.crma.2012.01.007.

[5]

S. Cordier, C. Lucas and J. D. D. Zabsonré, A two time-scale model for tidal bed-load transport, Communications in Mathematical Sciences, 10 (2012), 875-888. doi: 10.4310/CMS.2012.v10.n3.a8.

[6]

I. Faye, E. Frénod and D. Seck, Long term behavior of singularity perturbed parabolic degenerated equation, To appear in journal of non linear analysis and application.

[7]

D. Idier, D. Astruc and S. J. M. H. Hulcher, Influence of bed roughness on dune and megaripple generation, Geophysical Research Letters, 31 (2004), 1-5. doi: 10.1029/2004GL019969.

[8]

T. Kato, The Cauchy problem for quasi-linear symmetric system, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.

[9]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Appl. Math., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.

[10]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS Translation of Mathematical Monographs, 1968.

[11]

J. L. Lions, Remarques sur les équations différentielles ordinaires, Osaka Math. J., 15 (1963), 131-142.

[12]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.

[13]

L. C. Van Rijn, Handbook on Sediment Transport by Current and Waves, Tech. Report H461:12.1-12.27, Delft Hydraulics, 1989.

[14]

J. de D. Zabsonré, C. Lucas and E. Fernandez-Nieto, An energetically consistent viscous sedimentation model, Math. Model. Meth. Appl. Sci., 19 (2009), 477-499. doi: 10.1142/S0218202509003504.

show all references

References:
[1]

G. Allaire, Homogenization and Two-Scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.

[2]

P. Alliot, E. Frénod and V. Monbet, Modelling the coastal ocean over a time period of several weeks, Journal of Differential Equations, 248 (2010), 639-659. doi: 10.1016/j.jde.2009.11.004.

[3]

E. Audusse, F. Benkhadloun, J. Sainte-Marie and M. Seaid, Multilayer Saint-Venant equations over movable beds, Discrete and Continuous Dynamical Systems - B, 15 (2011), 917-934. doi: 10.3934/dcdsb.2011.15.917.

[4]

C. Berthon, S. Cordier, O. Delestre and M. H. Le, An analytical solution of shallow water system coupled to Exner equation, Comptes Rendus Mathématique, Elsevier, 350 (2012), 183-186. doi: 10.1016/j.crma.2012.01.007.

[5]

S. Cordier, C. Lucas and J. D. D. Zabsonré, A two time-scale model for tidal bed-load transport, Communications in Mathematical Sciences, 10 (2012), 875-888. doi: 10.4310/CMS.2012.v10.n3.a8.

[6]

I. Faye, E. Frénod and D. Seck, Long term behavior of singularity perturbed parabolic degenerated equation, To appear in journal of non linear analysis and application.

[7]

D. Idier, D. Astruc and S. J. M. H. Hulcher, Influence of bed roughness on dune and megaripple generation, Geophysical Research Letters, 31 (2004), 1-5. doi: 10.1029/2004GL019969.

[8]

T. Kato, The Cauchy problem for quasi-linear symmetric system, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.

[9]

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Appl. Math., 34 (1981), 481-524. doi: 10.1002/cpa.3160340405.

[10]

O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS Translation of Mathematical Monographs, 1968.

[11]

J. L. Lions, Remarques sur les équations différentielles ordinaires, Osaka Math. J., 15 (1963), 131-142.

[12]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.

[13]

L. C. Van Rijn, Handbook on Sediment Transport by Current and Waves, Tech. Report H461:12.1-12.27, Delft Hydraulics, 1989.

[14]

J. de D. Zabsonré, C. Lucas and E. Fernandez-Nieto, An energetically consistent viscous sedimentation model, Math. Model. Meth. Appl. Sci., 19 (2009), 477-499. doi: 10.1142/S0218202509003504.

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