Global solutions for the gravity water waves equation in dimension 3. (English) Zbl 1241.35003

Germain, Pierre; Masmoudi, Nader; Shatah, Jalal

Ann. Math. (2) 175, No. 2, 691-754 (2012).

This paper is concerned with the existence of global solutions for the gravity water waves equation, in the case of small data. The proof of existence result is based on the method of space-time resonances.

Reviewer: Valeriu Al. Sava (Paris)

MSC:

35A01	Existence problems for PDEs: global existence, local existence, non-existence
35L45	Initial value problems for first-order hyperbolic systems
35Q31	Euler equations
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction

Keywords:

gravity water waves; global existence; energy estimates; spaces-time resonances

References:

[1]	D. M. Ambrose and N. Masmoudi, ”Well-posedness of 3D vortex sheets with surface tension,” Commun. Math. Sci., vol. 5, iss. 2, pp. 391-430, 2007. · Zbl 1130.76016
[2]	D. M. Ambrose and N. Masmoudi, ”The zero surface tension limit of three-dimensional water waves,” Indiana Univ. Math. J., vol. 58, iss. 2, pp. 479-521, 2009. · Zbl 1172.35058
[3]	T. J. Beale, T. Y. Hou, and J. S. Lowengrub, ”Growth rates for the linearized motion of fluid interfaces away from equilibrium,” Comm. Pure Appl. Math., vol. 46, iss. 9, pp. 1269-1301, 1993. · Zbl 0796.76041
[4]	D. Christodoulou, ”Global solutions of nonlinear hyperbolic equations for small initial data,” Comm. Pure Appl. Math., vol. 39, iss. 2, pp. 267-282, 1986. · Zbl 0612.35090
[5]	<a href=’http://dx.doi.org/10.1002/1097-0312(200012)53:123.3.CO;2-H’ title=’Go to document’> D. Christodoulou and H. Lindblad, ”On the motion of the free surface of a liquid,” Comm. Pure Appl. Math., vol. 53, iss. 12, pp. 1536-1602, 2000. · Zbl 1031.35116
[6]	R. R. Coifman and Y. Meyer, Au delà des Opérateurs Pseudo-Différentiels, Paris: Société Mathématique de France, 1978, vol. 57. · Zbl 0483.35082
[7]	D. Coutand and S. Shkoller, ”Well-posedness of the free-surface incompressible Euler equations with or without surface tension,” J. Amer. Math. Soc., vol. 20, iss. 3, pp. 829-930, 2007. · Zbl 1123.35038
[8]	W. Craig, ”An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits,” Comm. Partial Differential Equations, vol. 10, iss. 8, pp. 787-1003, 1985. · Zbl 0577.76030
[9]	W. Craig and D. P. Nicholls, ”Traveling two and three dimensional capillary gravity water waves,” SIAM J. Math. Anal., vol. 32, iss. 2, pp. 323-359, 2000. · Zbl 0976.35049
[10]	G. B. Folland, Introduction to Partial Differential Equations, Second ed., Princeton, NJ: Princeton Univ. Press, 1995. · Zbl 0841.35001
[11]	P. Germain, N. Masmoudi, and J. Shatah, ”Global solutions for 3D quadratic Schrödinger equations,” Int. Math. Res. Not. IMRN, pp. 414-432, 2009. · Zbl 1156.35087
[12]	T. Iguchi, ”Well-posedness of the initial value problem for capillary-gravity waves,” Funkcial. Ekvac., vol. 44, iss. 2, pp. 219-241, 2001. · Zbl 1145.76328
[13]	S. Klainerman, ”Uniform decay estimates and the Lorentz invariance of the classical wave equation,” Comm. Pure Appl. Math., vol. 38, iss. 3, pp. 321-332, 1985. · Zbl 0635.35059
[14]	D. Lannes, ”Well-posedness of the water-waves equations,” J. Amer. Math. Soc., vol. 18, iss. 3, pp. 605-654, 2005. · Zbl 1069.35056
[15]	C. Muscalu, ”Paraproducts with flag singularities. I. A case study,” Rev. Mat. Iberoam., vol. 23, iss. 2, pp. 705-742, 2007. · Zbl 1213.42071
[16]	V. I. Nalimov, ”The Cauchy-Poisson problem,” Dinamika Splo\vsn. Sredy, iss. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami, pp. 104-210, 254, 1974.
[17]	M. Ogawa and A. Tani, ”Free boundary problem for an incompressible ideal fluid with surface tension,” Math. Models Methods Appl. Sci., vol. 12, iss. 12, pp. 1725-1740, 2002. · Zbl 1023.76007
[18]	G. Schneider and E. C. Wayne, ”The rigorous approximation of long-wavelength capillary-gravity waves,” Arch. Ration. Mech. Anal., vol. 162, iss. 3, pp. 247-285, 2002. · Zbl 1055.76006
[19]	J. Shatah, ”Normal forms and quadratic nonlinear Klein-Gordon equations,” Comm. Pure Appl. Math., vol. 38, iss. 5, pp. 685-696, 1985. · Zbl 0597.35101
[20]	J. Shatah and C. Zeng, ”Geometry and a priori estimates for free boundary problems of the Euler equation,” Comm. Pure Appl. Math., vol. 61, iss. 5, pp. 698-744, 2008. · Zbl 1174.76001
[21]	J. Shatah and C. Zeng, ”Local well-posedness for fluid interface problems,” Arch. Ration. Mech. Anal., vol. 199, iss. 2, pp. 653-705, 2011. · Zbl 1262.76034
[22]	E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton, NJ: Princeton Univ. Press, 1993, vol. 43. · Zbl 0821.42001
[23]	C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, New York: Springer-Verlag, 1999, vol. 139. · Zbl 0928.35157
[24]	E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton, N.J.: Princeton Univ. Press, 1971, vol. 32. · Zbl 0232.42007
[25]	S. Wu, ”Well-posedness in Sobolev spaces of the full water wave problem in \(2\)-D,” Invent. Math., vol. 130, iss. 1, pp. 39-72, 1997. · Zbl 0892.76009
[26]	S. Wu, ”Well-posedness in Sobolev spaces of the full water wave problem in 3-D,” J. Amer. Math. Soc., vol. 12, iss. 2, pp. 445-495, 1999. · Zbl 0921.76017
[27]	S. Wu, Almost global wellposedness of the 2-D full water wave problem, 2009. · Zbl 1181.35205
[28]	S. Wu, Global well-posedness of the 3-D full water wave problem, 2009. · Zbl 1181.35205
[29]	H. Yosihara, ”Gravity waves on the free surface of an incompressible perfect fluid of finite depth,” Publ. Res. Inst. Math. Sci., vol. 18, iss. 1, pp. 49-96, 1982. · Zbl 0493.76018
[30]	P. Zhang and Z. Zhang, ”On the free boundary problem of three-dimensional incompressible Euler equations,” Comm. Pure Appl. Math., vol. 61, 2008. · Zbl 1158.35107