## Large time existence for 3D water waves and asymptotics.(English)Zbl 1131.76012

Summary: We rigorously justify in three dimensions the main asymptotic models used in coastal oceanography, including: shallow-water equations, Boussinesq systems, Kadomtsev-Petviashvili approximation, Green-Naghdi equations, Serre approximation and full-dispersion model. We first introduce a nondimensionalized version of water wave equations which vary from shallow to deep water, and which involves four dimensionless parameters. Using a nonlocal energy adapted to the equations, we can prove a well-posedness theorem, uniformly with respect to all the parameters. Its validity ranges therefore from shallow to deep water, from small to large surface and bottom variations, and from fully to weakly transverse waves.
The physical regimes corresponding to the aforementioned models can therefore be studied as particular cases; it turns out that the existence time and energy bounds given by the theorem are always those needed to justify the asymptotic models. We can therefore derive and justify them in a systematic way.

### MSC:

 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics 35Q35 PDEs in connection with fluid mechanics

### Keywords:

asymptotic models; well-posedness; energy bounds
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### References:

 [1] Airy, G.B.: Tides and waves. Encyclopaedia Metropolitana, vol. 5, pp. 241–396. London (1845) [2] Alvarez-Samaniego, B., Lannes, D.: A Nash–Moser theorem for singular evolution equations. Application to the Serre and Green–Naghdi equations. Indiana Univ. Math. J., to appear · Zbl 1144.35007 [3] Ambrose, D., Masmoudi, N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58, 1287–1315 (2005) · Zbl 1086.76004 [4] Ben Youssef, W., Lannes, D.: The long wave limit for a general class of 2D quasilinear hyperbolic problems. Commun. Partial Differ. Equations 27, 979–1020 (2002) · Zbl 1072.35572 [5] Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178, 373–410 (2005) · Zbl 1108.76012 [6] 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 [7] Bona, J.L., Smith, R.: A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Philos. Soc. 79, 167–182 (1976) · Zbl 0332.76007 [8] 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 [9] Chazel, F.: Influence of topography on water waves. ESAIM: M2AN 41(4), 771–799 (2007) · Zbl 1144.76005 [10] Chen, M.: Equations for bi-directional waves over an uneven bottom. Math. Comput. Simul. 62, 3–9 (2003) · Zbl 1013.76014 [11] Choi, W.: Nonlinear evolution equations for two-dimensional surface waves in a fluid of finite depth. J. Fluid Mech. 295, 381–394 (1995) · Zbl 0920.76013 [12] Coutand, D., Shkoller, S.: Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20, 829–930 (2007) · Zbl 1123.35038 [13] Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg–de Vries scaling limits. Commun. Partial Differ. Equations 10, 787–1003 (1985) · Zbl 0577.76030 [14] Craig, W.: Nonstrictly hyperbolic nonlinear systems. Math. Ann. 277, 213–232 (1987) · Zbl 0614.35060 [15] Craig, W., Guyenne, P., Hammack, J., Henderson, D., Sulem, C.: Solitary water wave interactions. Phys. Fluids 18(5), 057106, 25pp. (2006) · Zbl 1185.76463 [16] Craig, W., Schanz, U., Sulem, C.: The modulational regime of three-dimensional water waves and the Davey–Stewartson system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, 615–667 (1997) · Zbl 0892.76008 [17] Craig, W., Sulem, C., Sulem, P.-L.: Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5, 497–522 (1992) · Zbl 0742.76012 [18] Dingemans, M.W.: Water wave propagation over uneven bottoms. Part 2: Non-Linear Wave Propagation. Adv. Ser. Ocean Eng., vol. 13. World Scientific, Singapore (1997) · Zbl 0908.76002 [19] Friedrichs, K.O.: On the derivation of the shallow water theory, Appendix to: Stoker, J.J.: The formulation of breakers and bores. Commun. Pure Appl. Math. 1, 1–87 (1948) [20] Gallay, T., Schneider, G.: KP description of unidirectional long waves. The model case. Proc. R. Soc. Edinb., Sect. A, Math. 131, 885–898 (2001) · Zbl 1015.76015 [21] Green, A.E., Laws, N., Naghdi, P. M.: On the theory of water wave. Proc. R. Soc. Lond., Ser. A 338, 43–55 (1974) · Zbl 0289.76010 [22] Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246 (1976) · Zbl 0351.76014 [23] Iguchi, T.: A long wave approximation for capillary-gravity waves and an effect of the bottom. Commun. Partial Differ. Equations 32, 37–85 (2007) · Zbl 1136.35081 [24] Iguchi, T.: A shallow water approximation for water waves. Preprint · Zbl 1421.76020 [25] Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys., Dokl. 15, 539–541 (1970) · Zbl 0217.25004 [26] Kano, T.: L’équation de Kadomtsev–Petviashvili approchant les ondes longues de surface de l’eau en écoulement trois-dimensionnel. Patterns and waves. Qualitative analysis of nonlinear differential equations. Stud. Math. Appl., vol. 18, 431–444. North-Holland, Amsterdam (1986) [27] Kano, T., Nishida, T.: Sur les ondes de surface de l’eau avec une justification mathématique des équations des ondes en eau peu profonde. J. Math. Kyoto Univ. 19, 335–370 (1979) · Zbl 0419.76013 [28] Kano, T., Nishida, T.: A mathematical justification for Korteweg–de Vries equation and Boussinesq equation of water surface waves. Osaka J. Math. 23, 389–413 (1986) · Zbl 0622.76021 [29] Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, 605–654 (2005) · Zbl 1069.35056 [30] Lannes, D.: Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators. J. Funct. Anal. 232, 495–539 (2006) · Zbl 1099.35191 [31] Bardos, C., Fursikov, A. (eds.): Instability in Models Connected with Fluid Flows II. Int. Math. Ser., vol. 7. Springer (2008) · Zbl 1130.76004 [32] Lannes, D., Saut, J.-C.: Weakly transverse Boussinesq systems and the KP approximation. Nonlinearity 19, 2853–2875 (2006) · Zbl 1122.35114 [33] Li, Y.A.: A shallow-water approximation to the full water wave problem. Commun. Pure Appl. Math. 59, 1225–1285 (2006) · Zbl 1169.76012 [34] Lindblad, H.: Well-posedness for the linearized motion of an incompressible liquid with free surface boundary. Commun. Pure Appl. Math. 56, 153–197 (2003) · Zbl 1025.35017 [35] Lindblad, H.: Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. (2) 162, 109–194 (2005) · Zbl 1095.35021 [36] Matsuno, Y.: Nonlinear evolution of surface gravity waves on fluid of finite depth. Phys. Rev. Lett. 69, 609–611 (1992) · Zbl 0968.76516 [37] Matsuno, Y.: Nonlinear evolution of surface gravity waves over an uneven bottom. J. Fluid Mech. 249, 121–133 (1993) · Zbl 0783.76015 [38] Nalimov, V.I.: The Cauchy–Poisson problem. Din. Splosh. Sredy 245, 104–210 (1974) · Zbl 0305.62048 [39] Nicholls, D., Reitich, F.: A new approach to analycity of Dirichlet–Neumann operators. Proc. R. Soc. Edinb., Sect. A 131, 1411–1433 (2001) · Zbl 1016.35030 [40] Ovsjannikov, L.V.: To the shallow water theory foundation. Arch. Mech. 26, 407–422 (1974) · Zbl 0283.76012 [41] Ovsjannikov, L.V.: Cauchy problem in a scale of Banach spaces and its application to the shallow water theory justification. In: Appl. Meth. Funct. Anal. Probl. Mech. (IUTAM/IMU-Symp., Marseille, 1975). Lect. Notes Math., vol. 503, 426–437 (1976) [42] Paumond, L.: A rigorous link between KP and a Benney-Luke equation. Differ. Integral Equ. 16, 1039–1064 (2003) · Zbl 1056.76014 [43] Schneider, G., Wayne, C.E.: The long-wave limit for the water wave problem. I: The case of zero surface tension. Commun. Pure Appl. Math. 53, 1475–1535 (2000) · Zbl 1034.76011 [44] Serre, F.: Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille Blanche 3, 374–388 (1953) [45] Shatah, J., Zeng, C.: Geometry and a priori estimates for free boundary problems of the Euler’s equation. Preprint (http://arxiv.org/abs/math.AP/0608428), Comm. Pure Appl. Math. (to appear) · Zbl 1174.76001 [46] Su, C.-H., Gardner, C.S.: Korteweg–de Vries equation and generalizations. III: Derivation of the Korteweg–de Vries equation and Burgers’ equation. J. Math. Phys. 10, 536–539 (1969) · Zbl 0283.35020 [47] Wright, J.D.: Corrections to the KdV approximation for water waves. SIAM J. Math. Anal. 37, 1161–1206 (2005) · Zbl 1093.76010 [48] 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 [49] Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445–495 (1999) · Zbl 0921.76017 [50] Yosihara, H.: Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18, 49–96 (1982) · Zbl 0493.76018 [51] Yosihara, H.: Capillary-gravity waves for an incompressible ideal fluid. J. Math. Kyoto Univ. 23, 649–694 (1983) · Zbl 0548.76018 [52] Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194 (1968)
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