We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy introduced by Bona et al. [Arch. Rational Mech. Anal. 178 (2005) 373-410], we derive new symetric asymptotic models for each regime. The solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones.
@article{M2AN_2007__41_4_771_0,
author = {Chazel, Florent},
title = {Influence of bottom topography on long water waves},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {41},
year = {2007},
pages = {771-799},
doi = {10.1051/m2an:2007041},
mrnumber = {2362914},
zbl = {pre05289496},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2007__41_4_771_0}
}
Chazel, Florent. Influence of bottom topography on long water waves. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) pp. 771-799. doi : 10.1051/m2an:2007041. http://gdmltest.u-ga.fr/item/M2AN_2007__41_4_771_0/
[1] and, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Savoirs Actuels, InterEditions, Paris, Editions du Centre National de la Recherche Scientifique (CNRS), Meudon (1991) p. 190. | MR 1172111 | Zbl 0791.47044
[2] and, Large time existence for 3D water-waves and asymptotics. Technical report (http://fr.arxiv.org/abs/math/0702015v1), Université Bordeaux I, IMB (2007). | MR 2372806 | Zbl 1131.76012
[3] and, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations. Preprint (http://arxiv.org/abs/math.AP/0701681v1), Indiana University Mathematical Journal (2007) (to appear). | MR 2400253 | Zbl pre05270592
[4] , and, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272 (1972) 47-78. | Zbl 0229.35013
[5] and, A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D 116 (2004) 191-224. | Zbl 0962.76515
[6] , and, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J. Nonlinear Sci. 12 (2002) 283-318. | Zbl 1022.35044
[7] , and, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: Nonlinear theory. Nonlinearity 17 (2004) 925-952. | Zbl 1059.35103
[8] , and, Long waves approximations for water waves. Arch. Rational Mech. Anal. 178 (2005) 373-410. | Zbl 1108.76012
[9] , Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C.R. Acad. Sci. Paris Sér. A-B 72 (1871) 755-759. | JFM 03.0486.01
[10] , Equations for bi-directional waves over an uneven bottom. Math. Comput. Simulation 62 (2003) 3-9. | Zbl 1013.76014
[11] , An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differential Equations 10 (1985) 787-1003. | Zbl 0577.76030
[12] , Water Wave Propagation over uneven bottoms. Part I: Linear Wave Propagation. Adanced Series on Ocean Engineering 13. World Scientific (1997). | Zbl 0908.76002
[13] , Water Wave Propagation over uneven bottoms. Part II: Non-linear Wave Propagation. Adanced Series on Ocean Engineering 13. World Scientific (1997). | Zbl 0908.76002
[14] and, A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78 (1976) 237-246. | Zbl 0351.76014
[15] , A long wave approximation for capillary-gravity waves and an effect of the bottom. Preprint (2005). | MR 2304142 | Zbl 1136.35081
[16] , A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves. Kyushu J. Math. 60 (2006) 267-303. | Zbl 1120.35078
[17] , Gravity Waves in Water of Finite Depth, Advances in Fluid Mechanics 10, in J.N. Hunt Ed., Computational Mechanics Publications (1997) 55-125. | Zbl 0900.76003
[18] , Sur le caractère bien posé des équations d'Euler avec surface libre. Séminaire EDP de l'École Polytechnique (2004), Exposé no. XIV. | Numdam
[19] , Well-posedness of the water-waves equations. J. Amer. Math. Soc. 18 (2005) 605-654. | Zbl 1069.35056
[20] and, Weakly transverse Boussinesq systems and the Kadomtsev-Petviashvili approximation. Nonlinearity 19 (2006) 2853-2875. | Zbl 1122.35114
[21] , and, A new form of the Boussinesq equations with improved linear dispersion characteristics (Part 1). Coastal Eng. 15 (1991) 371-388.
[22] , Small Viscosity and Boundary Layer Methods: Theory, Stability Analysis, and Applications. Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston-Basel-Berlin (2004). | MR 2151414 | Zbl 1133.35001
[23] and, A new approach to analyticity of Dirichlet-Neumann operators. Proc. Royal Soc. Edinburgh Sect. A 131 (2001) 1411-1433. | Zbl 1016.35030
[24] , The Cauchy-Poisson problem. (Russian) Dinamika Splošn. Sredy Vyp. 18, Dinamika Zidkost. so Svobod. Granicami 254 (1974) 104-210.
[25] , Alternative form of Boussinesq equations for nearshore wave propagation. J. Waterw. Port Coastal Eng. ASCE 119 (1993) 618-638.
[26] , Long waves on a beach. J. Fluid Mech. 27 (1967) 815-827. | Zbl 0163.21105
[27] and, The long-wave limit for the water-wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 162 (2002) 247-285. | Zbl 1055.76006
[28] and, A time-dependent numerical code for extended Boussinesq equations. J. Waterw. Port Coastal Ocean Engineering 120 (1995) 251-261.
[29] ,, and, A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear, unsteady waves. J. Fluid Mechanics 294 (1995) 71-92. | Zbl 0859.76009
[30] , Well-posedness in Sobolev spaces of the full water wave problem in -D. Invent. Math. 130 (1997) 39-72. | Zbl 0892.76009
[31] , Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Amer. Math. Soc. 12 (1999) 445-495. | Zbl 0921.76017
[32] , Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18 (1982) 49-96. | Zbl 0493.76018