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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 |
References:
[1] |
G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.
doi: 10.1137/0523084. |
[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. |
[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. |
[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. |
[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. |
[6] |
P. Blondeau, Mechanics of coastal forms,, Ann. Rev. Fluids Mech., 33 (2001), 339.
doi: 10.1146/annurev.fluid.33.1.339. |
[7] |
M. Bostan, Periodic solutions for evolution equations,, Elec. J. Diff. Equations. Monograph, 3 (2002), 1.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[22] |
J.-L. Lions, Remarques sur les équations différentielles ordinaires,, Osaka Math. J., 15 (1963), 131.
|
[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.
|
[26] |
G. Namah and J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinearparabolic equations,, Nonlinear Diff. Equ. Appl., 4 (1997), 521.
|
[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. |
[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. |
[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. |
[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.
|
[31] |
F. Petitta, Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data,, Elec. J. Diff. Equ., 2008 (2008), 1.
|
[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. |
[33] |
L. C. Van Rijn, "Handbook on Sediment Transport by Current and Waves,", Tech. Report H461:12.1-12.27, (1989), 1. Google Scholar |
show all references
References:
[1] |
G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.
doi: 10.1137/0523084. |
[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. |
[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. |
[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. |
[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. |
[6] |
P. Blondeau, Mechanics of coastal forms,, Ann. Rev. Fluids Mech., 33 (2001), 339.
doi: 10.1146/annurev.fluid.33.1.339. |
[7] |
M. Bostan, Periodic solutions for evolution equations,, Elec. J. Diff. Equations. Monograph, 3 (2002), 1.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[22] |
J.-L. Lions, Remarques sur les équations différentielles ordinaires,, Osaka Math. J., 15 (1963), 131.
|
[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.
|
[26] |
G. Namah and J.-M. Roquejoffre, Convergence to periodic fronts in a class of semilinearparabolic equations,, Nonlinear Diff. Equ. Appl., 4 (1997), 521.
|
[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. |
[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. |
[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. |
[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.
|
[31] |
F. Petitta, Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data,, Elec. J. Diff. Equ., 2008 (2008), 1.
|
[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. |
[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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