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

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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

Abstract
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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 & 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. 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. 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. 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, 350* (2012), 183. 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. 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. doi: 10.1029/2004GL019969.
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric system,, Arch. Ration. Mech. Anal., 58 (1975), 181. 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. 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.
[12]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043.
[13]	L. C. Van Rijn, Handbook on Sediment Transport by Current and Waves,, Tech. Report H461:12.1-12.27, (1989), 1.
[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. doi: 10.1142/S0218202509003504.

show all references

References:

[1]	G. Allaire, Homogenization and Two-Scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. 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. 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. 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, 350* (2012), 183. 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. 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. doi: 10.1029/2004GL019969.
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric system,, Arch. Ration. Mech. Anal., 58 (1975), 181. 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. 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.
[12]	G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608. doi: 10.1137/0520043.
[13]	L. C. Van Rijn, Handbook on Sediment Transport by Current and Waves,, Tech. Report H461:12.1-12.27, (1989), 1.
[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. doi: 10.1142/S0218202509003504.

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