Comptes Rendus
no. 15-16
Partial Differential Equations
Global solutions for the gravity water waves equation in dimension 3
[Solutions globales pour les équations des ondes de surface en dimension 3]

Présenté par : Gilles Lebeau

P. Germain1 ; Nader Masmoudi1 ; Jalal Shatah1
1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012-1185, USA
Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 897-902.

Nous montrons l'existence de solutions globales pour les équations des ondes de surface en dimension 3 avec gravité seulement, dans le cas de petites données initiales. La preuve combine des estimations d'énergie, qui donnent le contrôle de normes de type L2, avec des estimations dispersives, qui donnent la décroissance dans L∞. Ces estimations dispersives sont obtenues grâce à une analyse dans l'espace de Fourier, qui repose sur l'étude des résonances en temps et en espace.

We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data. The proof combines energy estimates, which yield control of L2 related norms, with dispersive estimates, which give decay in L∞. To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.05.005

P. Germain 1 ; Nader Masmoudi 1 ; Jalal Shatah 1

1 Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012-1185, USA 
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P. Germain; Nader Masmoudi; Jalal Shatah. Global solutions for the gravity water waves equation in dimension 3. Comptes Rendus. Mathématique, Volume 347 (2009) no. 15-16, pp. 897-902. doi : 10.1016/j.crma.2009.05.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.05.005/

[1] D.M. Ambrose; N. Masmoudi Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci., Volume 5 (2007) no. 2, pp. 391-430

[2] D.M. Ambrose, N. Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Math. J., in press

[3] J.T. Beale; T. Hou; J. Lowengrub Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math., Volume 46 (1993), pp. 1269-1301

[4] D. Christodoulou; H. Lindblad On the motion of the free surface of a liquid, Comm. Pure Appl. Math., Volume 53 (2000) no. 12, pp. 1536-1602

[5] R. Coifman; Y. Meyer Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978

[6] D. Coutand, S. Shkoller, Well posedness of the free-surface incompressible Euler equations with or without surface tension, Preprint

[7] W. Craig An existence theory for water waves and the Boussinesq and Korteweg–de Vries scaling limits, Comm. Partial Differential Equations, Volume 10 (1985) no. 8, pp. 787-1003

[8] P. Germain; N. Masmoudi; J. Shatah Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN, Volume 3 (2009), pp. 414-432

[9] T. Iguchi Well-posedness of the initial value problem for capillary-gravity waves, Funkcial. Ekvac., Volume 44 (2001) no. 2, pp. 219-241

[10] D. Lannes Well-posedness of the water-waves equations, J. Amer. Math. Soc., Volume 18 (2005), pp. 605-654

[11] C. Muscalu Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoamericana, Volume 23 (2007) no. 2, pp. 705-742

[12] V.I. Nalimov The Cauchy–Poisson problem, Dinamika Sploshn. Sredy Vyp. 18 Dinamika Zidkost. so Svobod. Granicami, Volume 254 (1974), pp. 104-210 (in Russian)

[13] J. Shatah; C. Zeng Geometry and a priori estimates for free boundary problems of the Euler's equation, Comm. Pure Appl. Math., Volume 51 (2008) no. 5, pp. 698-744

[14] C. Sulem; P.-L. Sulem The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999

[15] S. Wu Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., Volume 130 (1997) no. 1, pp. 39-72

[16] S. Wu Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., Volume 12 (1999) no. 2, pp. 445-495

[17] S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., in press

[18] H. Yosihara Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., Volume 23 (1983) no. 4, pp. 649-694

[19] P. Zhang, Z. Zhang, On the free boundary problem of 3-D incompressible Euler equations, Preprint

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  • Aksel Bergfeldt; Wolfgang Staubach On the regularity of multilinear Schrödinger integral operators, Analysis and Applications, Volume 21 (2023) no. 02, p. 385 | DOI:10.1142/s0219530522500099
  • Camil Muscalu; Yujia Zhai Five-linear singular integral estimates of Brascamp–Lieb-type, Analysis PDE, Volume 15 (2022) no. 4, p. 1011 | DOI:10.2140/apde.2022.15.1011
  • Salvador Rodríguez-López; David Rule; Wolfgang Staubach Global boundedness of a class of multilinear Fourier integral operators, Forum of Mathematics, Sigma, Volume 9 (2021) | DOI:10.1017/fms.2021.13
  • Bradley Isom; Dionyssios Mantzavinos; Seungly Oh; Atanas Stefanov Polynomial bound and nonlinear smoothing for the Benjamin-Ono equation on the circle, Journal of Differential Equations, Volume 297 (2021), p. 25 | DOI:10.1016/j.jde.2021.06.018
  • Diego Córdoba; Charles Fefferman Water Waves with or Without Surface Tension, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2018), p. 1329 | DOI:10.1007/978-3-319-13344-7_30
  • Wolf-Patrick Düll On the Mathematical Description of Time-Dependent Surface Water Waves, Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 120 (2018) no. 2, p. 117 | DOI:10.1365/s13291-017-0173-6
  • Tarek Elgindi; Donghyun Lee Uniform regularity for free-boundary Navier–Stokes equations with surface tension, Journal of Hyperbolic Differential Equations, Volume 15 (2018) no. 01, p. 37 | DOI:10.1142/s0219891618500030
  • Yu Mei; Yong Wang; Zhouping Xin Uniform regularity for the free surface compressible Navier–Stokes equations with or without surface tension, Mathematical Models and Methods in Applied Sciences, Volume 28 (2018) no. 02, p. 259 | DOI:10.1142/s0218202518500082
  • Benjamin Harrop-Griffiths; Jeremy L Marzuola Small data global solutions for the Camassa–Choi equations, Nonlinearity, Volume 31 (2018) no. 5, p. 1868 | DOI:10.1088/1361-6544/aaa7b6
  • Nader Masmoudi; Frederic Rousset Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations, Archive for Rational Mechanics and Analysis, Volume 223 (2017) no. 1, p. 301 | DOI:10.1007/s00205-016-1036-5
  • Xianpeng Hu; Nader Masmoudi Global Solutions to Repulsive Hookean Elastodynamics, Archive for Rational Mechanics and Analysis, Volume 223 (2017) no. 1, p. 543 | DOI:10.1007/s00205-016-1039-2
  • Donghyun Lee Uniform Estimate of Viscous Free-Boundary Magnetohydrodynamics with Zero Vacuum Magnetic Field, SIAM Journal on Mathematical Analysis, Volume 49 (2017) no. 4, p. 2710 | DOI:10.1137/16m1089794
  • Daniel Coutand; Steve Shkoller On the Impossibility of Finite-Time Splash Singularities for Vortex Sheets, Archive for Rational Mechanics and Analysis, Volume 221 (2016) no. 2, p. 987 | DOI:10.1007/s00205-016-0977-z
  • Diego Córdoba; Charles Fefferman Water Waves With or Without Surface Tension, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (2016), p. 1 | DOI:10.1007/978-3-319-10151-4_30-1
  • David M. Ambrose Vortex Sheet Formulations and Initial Value Problems: Analysis and Computing, Lectures on the Theory of Water Waves (2016), p. 140 | DOI:10.1017/cbo9781316411155.009
  • Daniel Coutand; Steve Shkoller On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations, Communications in Mathematical Physics, Volume 325 (2014) no. 1, p. 143 | DOI:10.1007/s00220-013-1855-2
  • Yan Guo; Ian Tice Local well-posedness of the viscous surface wave problem without surface tension, Analysis PDE, Volume 6 (2013) no. 2, p. 287 | DOI:10.2140/apde.2013.6.287
  • Yan Guo; Ian Tice Decay of viscous surface waves without surface tension in horizontally infinite domains, Analysis PDE, Volume 6 (2013) no. 6, p. 1429 | DOI:10.2140/apde.2013.6.1429
  • Angel Castro; Diego Córdoba; Charles Fefferman; Francisco Gancedo; Javier Gómez-Serrano Finite time singularities for the free boundary incompressible Euler equations, Annals of Mathematics, Volume 178 (2013) no. 3, p. 1061 | DOI:10.4007/annals.2013.178.3.6
  • Yan Guo; Ian Tice Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension, Archive for Rational Mechanics and Analysis, Volume 207 (2013) no. 2, p. 459 | DOI:10.1007/s00205-012-0570-z
  • Fabio Pusateri; Jalal Shatah Space‐Time Resonances and the Null Condition for First‐Order Systems of Wave Equations, Communications on Pure and Applied Mathematics, Volume 66 (2013) no. 10, p. 1495 | DOI:10.1002/cpa.21461
  • Nader Masmoudi; Kenji Nakanishi Multifrequency NLS Scaling for a Model Equation of Gravity‐Capillary Waves, Communications on Pure and Applied Mathematics, Volume 66 (2013) no. 8, p. 1202 | DOI:10.1002/cpa.21464
  • Oana Pocovnicu First and Second Order Approximations for a Nonlinear Wave Equation, Journal of Dynamics and Differential Equations, Volume 25 (2013) no. 2, p. 305 | DOI:10.1007/s10884-013-9286-5
  • Wolf‐Patrick DÜll Validity of the Korteweg–de Vries approximation for the two‐dimensional water wave problem in the arc length formulation, Communications on Pure and Applied Mathematics, Volume 65 (2012) no. 3, p. 381 | DOI:10.1002/cpa.21381
  • P. Germain; N. Masmoudi; J. Shatah Global solutions for 2D quadratic Schrödinger equations, Journal de Mathématiques Pures et Appliquées, Volume 97 (2012) no. 5, p. 505 | DOI:10.1016/j.matpur.2011.09.008
  • T. Alazard; N. Burq; C. Zuily Low regularity Cauchy theory for the water-waves problem: canals and swimming pools, Journées équations aux dérivées partielles (2012), p. 1 | DOI:10.5802/jedp.75
  • Angel Castro; Diego Córdoba; Charles L. Fefferman; Francisco Gancedo; Javier Gómez-Serrano Splash singularity for water waves, Proceedings of the National Academy of Sciences, Volume 109 (2012) no. 3, p. 733 | DOI:10.1073/pnas.1115948108
  • Anatoli V. Babin; Alexei A. Ilyin; Edriss S. Titi On the regularization mechanism for the periodic Korteweg–de Vries equation, Communications on Pure and Applied Mathematics, Volume 64 (2011) no. 5, p. 591 | DOI:10.1002/cpa.20356
  • Sijue Wu Global wellposedness of the 3-D full water wave problem, Inventiones mathematicae, Volume 184 (2011) no. 1, p. 125 | DOI:10.1007/s00222-010-0288-1
  • Gregory R. Baker; Chao Xie Singularities in the complex physical plane for deep water waves, Journal of Fluid Mechanics, Volume 685 (2011), p. 83 | DOI:10.1017/jfm.2011.283
  • Angel Castro; Diego Córdoba; Charles L. Fefferman; Francisco Gancedo; María López-Fernández Turning waves and breakdown for incompressible flows, Proceedings of the National Academy of Sciences, Volume 108 (2011) no. 12, p. 4754 | DOI:10.1073/pnas.1101518108

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