Bona, Jerry L.; Colin, Thierry; Lannes, David Long wave approximations for water waves. (English) Zbl 1108.76012 Arch. Ration. Mech. Anal. 178, No. 3, 373-410 (2005). Summary: In this paper, we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions. These systems are symmetric and conservative. Rigorous convergence results are provided showing that solutions of the complete free-surface Euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large. Using this result as a tool, a rigorous justification of all the two-dimensional, approximate systems recently put forward and analysed by Bona, Chen and Saut is obtained. In the two-dimensional context, our methods also allows a significant improvement of the convergence estimate obtained by Schneider and Wayne in their justification of the decoupled Korteweg-de Vries approximation of the two-dimensional Euler equations. It also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales. Results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic. Cited in 5 ReviewsCited in 130 Documents MSC: 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35Q35 PDEs in connection with fluid mechanics 35Q53 KdV equations (Korteweg-de Vries equations) 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M., Wu, J.: Comparison between the BBM equation and a Boussinesq system. Preprint 2003 · Zbl 1104.35039 [2] Albert, J.P., Bona, J.L.: Comparisons between model equations for long waves. J. Nonlinear Sci. 1, 345–374 (1991) · Zbl 0791.35123 · doi:10.1007/BF01238818 [3] Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Royal Soc. London Ser. A 272, 47–78 (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032 [4] Ben Youssef, W., Colin, T.: Rigorous derivation of Korteweg-de Vries-type systems from a general class of nonlinear hyperbolic systems. M2AN Math. Model. Numer. Anal. 34, 873–911 (2000) · Zbl 0962.35152 [5] Boczar-Karakiewicz, B., Bona, J.L., Romańczyk, W., Thornton, E.: Sand bars at Duck, NC, USA. Observations and model predictions. In the proceedings of the conference Coastal Sediments ’99 (held in New York) American Soc. Civil Engineers: New York, 491–504 [6] Boczar-Karakiewicz, B., Bona, J.L., Romańczyk, W., Thornton, E.: Seasonal and interseasonal variability of sand bars at NC, USA. Observations and model predictions. Submitted [7] Bona, J.L., Chen, M.: A Boussinesq system for the two-way propagation of nonlinear dispersive waves. Physica D 116, 191–224 (1998) · Zbl 0962.76515 · doi:10.1016/S0167-2789(97)00249-2 [8] Bona, J.L., Chen, M., Saut, J.C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002) · Zbl 1022.35044 [9] Bona, J.L., Chen, M., Saut, J.C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. Nonlinear theory. Nonlinearity 17, 925–952 (2004) · Zbl 1059.35103 [10] Bona, J.L., Pritchard, W.G., Scott, L.R.: An evaluation of a model equation for water waves. Philos. Trans. Royal Soc. London, Ser. A 302, 457–510 (1981) · Zbl 0497.76023 · doi:10.1098/rsta.1981.0178 [11] Bona, J.L., Pritchard, W.G., Scott, L.R.: A comparison of solutions of two model equations for long waves. In: Fluid Dynamics in Astrophysics and Geophysics (ed. N. Lebovitz), vol. 20 of Lectures in Appl. Math., American Math. Soc., Providence, R.I.: 235–267, 1983 · Zbl 0534.76024 [12] Bona, J.L., Smith, R.: A model for the two-way propagation of water waves in a channel. Math. Proc. Cambridge Philos. Soc 79, 167–182 (1976) · Zbl 0332.76007 · doi:10.1017/S030500410005218X [13] Boussinesq. M.J.: 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, 755–759 (1871) · JFM 03.0486.01 [14] Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Comm. Partial Differential Equations 10, 787–1003 (1985) · Zbl 0577.76030 · doi:10.1080/03605308508820396 [15] Craig, W., Schantz, U., Sulem, C.: The modulational regime of three-dimensional water waves and the Davey-Stewartson system. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 615–667 (1997) · Zbl 0892.76008 · doi:10.1016/S0294-1449(97)80128-X [16] Craig, W., Sulem, C., Sulem, P.L.: Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5, 497–522 (1992) · Zbl 0742.76012 · doi:10.1088/0951-7715/5/2/009 [17] Lannes, D.: Dispersive effects for nonlinear geometrical optics with rectification. Asymptot. Anal. 18, 111–146 (1998) · Zbl 0931.35091 [18] Lannes, D.: Secular growth estimates for hyperbolic systems. J. Differential Equations 190, 466–503 (2003) · Zbl 1052.35119 · doi:10.1016/S0022-0396(02)00174-2 [19] Lannes, D.: Well-posedness of the water-waves equations. To appear in the J. American Math. Soc. · Zbl 1069.35056 [20] Lannes, D.: Sur le caractère bien posé des équations d’Euler avec surface libre. Séminaire EDP de l’Ecole Polytechnique (2004), Exposé no. XIV [21] Nalimov, V.I.: The Cauchy-Poisson problem. Dinamika Splošn. Sredy (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami) 254, 104–210 (1974) · Zbl 0305.62048 [22] Nicholls, D.P., Reitich, F.: A new approach to analyticity of Dirichlet-Neumann operators. Proc. Royal Soc. Edinburgh Sect. A 131, 1411–1433 (2001) · Zbl 1016.35030 · doi:10.1017/S0308210500001463 [23] Schneider, G., Wayne, C.E.: The long-wave limit for the water wave problem. I. The case of zero surface tension. Comm. Pure Appl. Math. 53, 1475–1535 (2000) · Zbl 1034.76011 [24] Wayne, C.E., Wright, J.D.: Higher order corrections to the KdV approximation for a Boussinesq equation. SIAM J. Appl. Dyn. Systems 1, 271–302 (2002) · Zbl 1088.76007 · doi:10.1137/S1111111102411298 [25] Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997) · Zbl 0892.76009 · doi:10.1007/s002220050177 [26] Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. American Math. Soc. 12, 445–495 (1999) · Zbl 0921.76017 · doi:10.1090/S0894-0347-99-00290-8 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.