Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment

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July 2011, 29(3): 1001-1030. doi: 10.3934/dcds.2011.29.1001

Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment

Ibrahima Faye ^1, , Emmanuel Frénod ^2, and Diaraf Seck ^3,

1.	Université de Bambey, BP 30 Bambey, Ecole Doctorale de Mathématiques et Informatique, Laboratoire de Mathématiques de la Décision et d'Analyse Numérique, F.A.S.E.G/F.S.T, Senegal
2.	Université Européenne de Bretagne, Lab-STICC (UMR CNRS 3192), Université de Bretagne-Sud, Centre Yves Coppens, Campus de Tohannic, F-56017, Vannes, France
3.	Université Cheikh Anta Diop de Dakar, BP 16 889, Dakar-Fann. E.D. de Mathématiques et Informatique, Laboratoire de Mathématiques de la Décision et d'Analyse Numérique, F.A.S.E.G/F.S.T, Senegal

Received January 2010 Revised July 2010 Published November 2010

Abstract
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In this paper we build models for short-term, mean-term and long-term dynamics of dune and megariple morphodynamics. They are models that are degenerated parabolic equations which are, moreover, singularly perturbed. We, then give an existence and uniqueness result for the short-term and mean-term models. This result is based on a time-space periodic solution existence result for degenerated parabolic equation that we set out. Finally the short-term model is homogenized.

Keywords: Asymptotic Expansion, Long Time Behavior, Space-Time Periodic Solutions, Dune and Megaripple Morphodynamics, Asymptotic Analysis, Homogenization, Degenerated Parabolic Equation, Modeling Coastal Zone Phenomena..

Mathematics Subject Classification: Primary: 35K65, 35B25, 35B10 ; Secondary: 92F05, 86A6.

Citation: Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1001-1030. doi: 10.3934/dcds.2011.29.1001

References:

[1]	G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar
[2]	R. A. Bagnold, The movement of desert sand,, Proceedings of the Royal Society of London A, 157 (1936), 594. doi: 10.1098/rspa.1936.0218. Google Scholar
[3]	G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasilinear parabolic equations,, SIAM J. Math. Anal., 32 (2001), 1311. doi: 10.1137/S0036141000369344. Google Scholar
[4]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodicity fragmented environment model: I-species persistence,, J. Math Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3. Google Scholar
[5]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodicity fragmented environment model: Ii-biological invasions and pulsating travelling fronts,, J. Math Pures Appl., 84 (2005), 1101. doi: 10.1016/j.matpur.2004.10.006. Google Scholar
[6]	P. Blondeau, Mechanics of coastal forms,, Ann. Rev. Fluids Mech., 33 (2001), 339. doi: 10.1146/annurev.fluid.33.1.339. Google Scholar
[7]	M. Bostan, Periodic solutions for evolution equations,, Elec. J. Diff. Equations. Monograph, 3 (2002), 1. Google Scholar
[8]	F. Da Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions,, J. Math. Anal. Appl., 339 (2008), 384. doi: 10.1016/j.jmaa.2007.06.052. Google Scholar
[9]	G. P. Dawson, B. Johns and R. L. Soulsby, A numerical model of shallow-water flow over topography,, in, 35 (1983), 267. doi: 10.1016/S0422-9894(08)70504-X. Google Scholar
[10]	H. J. De Vriend, "Steady Flow in Shallow Channel Bends,", Ph.D. thesis, (1981). Google Scholar
[11]	F. Engelund and E. Hansen, "Investigation of Flow in Alluvial Streams,", Tech. Report 9, (1966). Google Scholar
[12]	B. W. Flemming, The role of grain size, water depth and flow velocity as scaling factors controlling the size of subaqueous dunes,, Marine Sandwave Dynamics, (2000), 23. Google Scholar
[13]	E. Frénod, P. A. Raviart and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field,, J. Math. Pures et Appl., 80 (2001), 815. doi: 10.1016/S0021-7824(01)01215-6. Google Scholar
[14]	P. E. Gadd, W. Lavelle and D. J. P. Swift, Estimates of sand transport on the New York shelf using near-bottom current meter observations,, J. Sed. Petrol., 48 (1978), 239. Google Scholar
[15]	A. Hansbo, Error estimates for the numerical solution of a time-periodic linear parabolic problem,, BIT, 31 (1991), 664. doi: 10.1007/BF01933180. Google Scholar
[16]	D. Idier, "Dunes et Bancs de Sables du Plateau Continental: Observations in-situ et Modélisation Numérique,", Ph.D. thesis, (2002). Google Scholar
[17]	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. Google Scholar
[18]	B. Johns, R. Soulsby and T. Chesher, The modelling of sand waves evolution resulting from suspended and bed load transport of sediment,, J. Hydraul. Reseach, 28 (1990), 355. doi: 10.1080/00221689009499075. Google Scholar
[19]	J. Kennedy, The formation of sediment ripples, dunes and antidunes,, Ann. Rev. Fluids Mech., 1 (1969), 147. doi: 10.1146/annurev.fl.01.010169.001051. Google Scholar
[20]	M. Kono, Remarks on periodic solutions of linear parabolic differential equations of the second order,, Proc. Japan Acad., 42 (1966), 5. doi: 10.3792/pja/1195522166. Google Scholar
[21]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", AMS Translation of Mathematical Monographs \textbf{23}, 23 (1968). Google Scholar
[22]	J.-L. Lions, Remarques sur les équations différentielles ordinaires,, Osaka Math. J., 15 (1963), 131. Google Scholar
[23]	E. Meyer-Peter and R. Müller, Formulas for bed-load transport,, The Second Meeting of the International Association for Hydraulic Structures, (1948), 39. Google Scholar
[24]	G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion,, preprint., (). Google Scholar
[25]	G. Nadin, Reaction-diffusion equations in space-time periodic media,, C. R. Acad. Sci. Paris Ser. I, 345 (2007), 489. Google Scholar
[26]	G. Namah and J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinearparabolic equations,, Nonlinear Diff. Equ. Appl., 4 (1997), 521. Google Scholar
[27]	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. Google Scholar
[28]	J. R. Norris, Long-time behaviour of heat flow: Global estimates and exact asymptotics,, Arch. Rat. Mech. Anal., 140 (1997), 161. doi: 10.1007/s002050050063. Google Scholar
[29]	E. Pardoux, Homogenization of linear and semilinear second order parabolic pdes with periodic coefficients: A probolist approach,, J. Funct. Anal., 167 (1999), 498. doi: 10.1006/jfan.1999.3441. Google Scholar
[30]	D. G. Park and H. Tanabe, On the asymptotic behavior of solutions of linear parabolic equations in $l^1$ space,, Annali Delle Scuola Normale superiore di Pisa Classe di Scienze, 14 (1987), 587. Google Scholar
[31]	F. Petitta, Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data,, Elec. J. Diff. Equ., 2008 (2008), 1. Google Scholar
[32]	H. Tanabe, Convergence to a stationary state of the solution of some kind of differential equations in a banach space,, Proc. Japan Acad., 37 (1961), 127. doi: 10.3792/pja/1195523776. Google Scholar
[33]	L. C. Van Rijn, "Handbook on Sediment Transport by Current and Waves,", Tech. Report H461:12.1-12.27, (1989), 1. Google Scholar

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References:

[1]	G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar
[2]	R. A. Bagnold, The movement of desert sand,, Proceedings of the Royal Society of London A, 157 (1936), 594. doi: 10.1098/rspa.1936.0218. Google Scholar
[3]	G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasilinear parabolic equations,, SIAM J. Math. Anal., 32 (2001), 1311. doi: 10.1137/S0036141000369344. Google Scholar
[4]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodicity fragmented environment model: I-species persistence,, J. Math Biol., 51 (2005), 75. doi: 10.1007/s00285-004-0313-3. Google Scholar
[5]	H. Berestycki, F. Hamel and L. Roques, Analysis of the periodicity fragmented environment model: Ii-biological invasions and pulsating travelling fronts,, J. Math Pures Appl., 84 (2005), 1101. doi: 10.1016/j.matpur.2004.10.006. Google Scholar
[6]	P. Blondeau, Mechanics of coastal forms,, Ann. Rev. Fluids Mech., 33 (2001), 339. doi: 10.1146/annurev.fluid.33.1.339. Google Scholar
[7]	M. Bostan, Periodic solutions for evolution equations,, Elec. J. Diff. Equations. Monograph, 3 (2002), 1. Google Scholar
[8]	F. Da Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions,, J. Math. Anal. Appl., 339 (2008), 384. doi: 10.1016/j.jmaa.2007.06.052. Google Scholar
[9]	G. P. Dawson, B. Johns and R. L. Soulsby, A numerical model of shallow-water flow over topography,, in, 35 (1983), 267. doi: 10.1016/S0422-9894(08)70504-X. Google Scholar
[10]	H. J. De Vriend, "Steady Flow in Shallow Channel Bends,", Ph.D. thesis, (1981). Google Scholar
[11]	F. Engelund and E. Hansen, "Investigation of Flow in Alluvial Streams,", Tech. Report 9, (1966). Google Scholar
[12]	B. W. Flemming, The role of grain size, water depth and flow velocity as scaling factors controlling the size of subaqueous dunes,, Marine Sandwave Dynamics, (2000), 23. Google Scholar
[13]	E. Frénod, P. A. Raviart and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field,, J. Math. Pures et Appl., 80 (2001), 815. doi: 10.1016/S0021-7824(01)01215-6. Google Scholar
[14]	P. E. Gadd, W. Lavelle and D. J. P. Swift, Estimates of sand transport on the New York shelf using near-bottom current meter observations,, J. Sed. Petrol., 48 (1978), 239. Google Scholar
[15]	A. Hansbo, Error estimates for the numerical solution of a time-periodic linear parabolic problem,, BIT, 31 (1991), 664. doi: 10.1007/BF01933180. Google Scholar
[16]	D. Idier, "Dunes et Bancs de Sables du Plateau Continental: Observations in-situ et Modélisation Numérique,", Ph.D. thesis, (2002). Google Scholar
[17]	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. Google Scholar
[18]	B. Johns, R. Soulsby and T. Chesher, The modelling of sand waves evolution resulting from suspended and bed load transport of sediment,, J. Hydraul. Reseach, 28 (1990), 355. doi: 10.1080/00221689009499075. Google Scholar
[19]	J. Kennedy, The formation of sediment ripples, dunes and antidunes,, Ann. Rev. Fluids Mech., 1 (1969), 147. doi: 10.1146/annurev.fl.01.010169.001051. Google Scholar
[20]	M. Kono, Remarks on periodic solutions of linear parabolic differential equations of the second order,, Proc. Japan Acad., 42 (1966), 5. doi: 10.3792/pja/1195522166. Google Scholar
[21]	O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasi-Linear Equations of Parabolic Type,", AMS Translation of Mathematical Monographs \textbf{23}, 23 (1968). Google Scholar
[22]	J.-L. Lions, Remarques sur les équations différentielles ordinaires,, Osaka Math. J., 15 (1963), 131. Google Scholar
[23]	E. Meyer-Peter and R. Müller, Formulas for bed-load transport,, The Second Meeting of the International Association for Hydraulic Structures, (1948), 39. Google Scholar
[24]	G. Nadin, Existence and uniqueness of the solution of a space-time periodic reaction-diffusion,, preprint., (). Google Scholar
[25]	G. Nadin, Reaction-diffusion equations in space-time periodic media,, C. R. Acad. Sci. Paris Ser. I, 345 (2007), 489. Google Scholar
[26]	G. Namah and J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinearparabolic equations,, Nonlinear Diff. Equ. Appl., 4 (1997), 521. Google Scholar
[27]	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. Google Scholar
[28]	J. R. Norris, Long-time behaviour of heat flow: Global estimates and exact asymptotics,, Arch. Rat. Mech. Anal., 140 (1997), 161. doi: 10.1007/s002050050063. Google Scholar
[29]	E. Pardoux, Homogenization of linear and semilinear second order parabolic pdes with periodic coefficients: A probolist approach,, J. Funct. Anal., 167 (1999), 498. doi: 10.1006/jfan.1999.3441. Google Scholar
[30]	D. G. Park and H. Tanabe, On the asymptotic behavior of solutions of linear parabolic equations in $l^1$ space,, Annali Delle Scuola Normale superiore di Pisa Classe di Scienze, 14 (1987), 587. Google Scholar
[31]	F. Petitta, Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data,, Elec. J. Diff. Equ., 2008 (2008), 1. Google Scholar
[32]	H. Tanabe, Convergence to a stationary state of the solution of some kind of differential equations in a banach space,, Proc. Japan Acad., 37 (1961), 127. doi: 10.3792/pja/1195523776. Google Scholar
[33]	L. C. Van Rijn, "Handbook on Sediment Transport by Current and Waves,", Tech. Report H461:12.1-12.27, (1989), 1. Google Scholar

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