Abstract
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.
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Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M., Wu, J.: Comparison between the BBM equation and a Boussinesq system. Preprint 2003
Albert, J.P., Bona, J.L.: Comparisons between model equations for long waves. J. Nonlinear Sci. 1, 345–374 (1991)
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)
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)
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
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
Bona, J.L., Chen, M.: A Boussinesq system for the two-way propagation of nonlinear dispersive waves. Physica D 116, 191–224 (1998)
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)
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)
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)
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
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)
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)
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)
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)
Craig, W., Sulem, C., Sulem, P.L.: Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5, 497–522 (1992)
Lannes, D.: Dispersive effects for nonlinear geometrical optics with rectification. Asymptot. Anal. 18, 111–146 (1998)
Lannes, D.: Secular growth estimates for hyperbolic systems. J. Differential Equations 190, 466–503 (2003)
Lannes, D.: Well-posedness of the water-waves equations. To appear in the J. American Math. Soc.
Lannes, D.: Sur le caractère bien posé des équations d'Euler avec surface libre. Séminaire EDP de l'Ecole Polytechnique (2004), Exposé no. XIV
Nalimov, V.I.: The Cauchy-Poisson problem. Dinamika Splošn. Sredy (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami) 254, 104–210 (1974)
Nicholls, D.P., Reitich, F.: A new approach to analyticity of Dirichlet-Neumann operators. Proc. Royal Soc. Edinburgh Sect. A 131, 1411–1433 (2001)
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)
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)
Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997)
Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. American Math. Soc. 12, 445–495 (1999)
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Communicated by Y. Brenier
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Bona, J., Colin, T. & Lannes, D. Long Wave Approximations for Water Waves. Arch. Rational Mech. Anal. 178, 373–410 (2005). https://doi.org/10.1007/s00205-005-0378-1
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DOI: https://doi.org/10.1007/s00205-005-0378-1