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Hamiltonian Paradifferential Birkhoff Normal Form for Water Waves

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Abstract

We present the almost global in time existence result in [13] of small amplitude space periodic solutions of the 1D gravity-capillary water waves equations with constant vorticity and we describe the ideas of proof. This is based on a novel Hamiltonian paradifferential Birkhoff normal form approach for quasi-linear PDEs.

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Change history

  • 29 October 2023

    The numbering issue has been changed to 4-5.

References

  1. Alazard, T. and Baldi, P., Gravity Capillary Standing Water Waves, Arch. Ration. Mech. Anal., 2015, vol. 217, no. 3, pp. 741–830.

    MathSciNet  MATH  Google Scholar 

  2. Alazard, T., Burq, N., and Zuily, C., On the Water-Wave Equations with Surface Tension, Duke Math. J., 2011, vol. 158, no. 3, pp. 413–499.

    MathSciNet  MATH  Google Scholar 

  3. Alazard, T., Burq, N., and Zuily, C., On the Cauchy Problem for Gravity Water Waves, Invent. Math., 2014, vol. 198, no. 1, pp. 71–163.

    MathSciNet  MATH  Google Scholar 

  4. Alazard, T. and Delort, J.-M., Global Solutions and Asymptotic Behavior for Two Dimensional Gravity Water Waves, Ann. Sci. Éc. Norm. Supér. (4), 2015, vol. 48, no. 5, pp. 1149–1238.

    MathSciNet  MATH  Google Scholar 

  5. Alazard, T. and Métivier, G., Paralinearization of the Dirichlet to Neumann Operator, and Regularity of Three-Dimensional Water Waves, Comm. Partial Differential Equations, 2009, vol. 34, no. 10–12, pp. 1632–1704.

    MathSciNet  MATH  Google Scholar 

  6. Bambusi, D., Birkhoff Normal Form for Some Nonlinear PDEs, Comm. Math. Phys., 2003, vol. 234, no. 2, pp. 253–285.

    MathSciNet  MATH  Google Scholar 

  7. Bambusi, D., Asymptotic Stability of Ground States in Some Hamiltonian PDEs with Symmetry, Comm. Math. Phys., 2013, vol. 320, no. 2, pp. 499–542.

    MathSciNet  MATH  Google Scholar 

  8. Bambusi, D. and Maspero, A., Birkhoff Coordinates for the Toda Lattice in the Limit of Infinitely Many Particles with an Application to FPU, J. Funct. Anal., 2016, vol. 270, no. 5, pp. 1818–1887.

    MathSciNet  MATH  Google Scholar 

  9. Bambusi, D. and Maspero, A., Freezing of Energy of a Soliton in an External Potential, Comm. Math. Phys., 2016, vol. 344, no. 1, pp. 155–191.

    MathSciNet  MATH  Google Scholar 

  10. Berti, M. and Delort, J.-M., Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle, Lecture Notes of the Unione Matematica Italiana, vol. 24, Cham: Springer, 2018, https://link.springer.com/book/10.1007/978-3-319-99486-4.

    MATH  Google Scholar 

  11. Berti, M., Feola, R., and Franzoi, L., Quadratic Life Span of Periodic Gravity-Capillary Water Waves, Water Waves, 2021, vol. 3, no. 1, pp. 85–115.

    MathSciNet  MATH  Google Scholar 

  12. Berti, M., Feola, R., and Pusateri, F., Birkhoff Normal Form and Long Time Existence for Periodic Gravity Water Waves, Comm. Pure Appl. Math., 2023, vol. 76, no. 7, pp. 1416–1494.

    MathSciNet  MATH  Google Scholar 

  13. Berti, M., Maspero, A., and Murgante, F., Hamiltonian Birkhoff Normal Form for Gravity-Capillary Water Waves with Constant Vorticity: Almost Global Existence, arXiv:2212.12255 (2022).

  14. Berti, M., Franzoi, L., and Maspero, A., Traveling Quasi-Periodic Water Waves with Constant Vorticity, Arch. Ration. Mech. Anal., 2021, vol. 240, no. 1, pp. 99–202.

    MathSciNet  MATH  Google Scholar 

  15. Berti, M. and Montalto, R., Quasi-Periodic Standing Wave Solutions of Gravity-Capillary Water Waves, Mem. Amer. Math. Soc., vol. 263, no. 1273, Providence, R.I.: AMS, 2020.

    MATH  Google Scholar 

  16. Beyer, K. and Günther, M., On the Cauchy Problem for a Capillary Drop: 1. Irrotational Motion, Math. Methods Appl. Sci., 1998, vol. 21, no. 12, pp. 1149–1183.

    MathSciNet  MATH  Google Scholar 

  17. Castro, A., Córdoba, D., Fefferman, Ch., Gancedo, F., Gómez-Serrano, J., Finite Time Singularities for the Free Boundary Incompressible Euler Equations, Ann. of Math. (2), 2013, vol. 178, no. 3, pp. 1061–1134.

    MathSciNet  MATH  Google Scholar 

  18. Constantin, A., Ivanov, R. I., and Prodanov, E. M., Nearly-Hamiltonian Structure for Water Waves with Constant Vorticity, J. Math. Fluid Mech., 2008, vol. 10, no. 2, pp. 224–237.

    MathSciNet  MATH  Google Scholar 

  19. Craig, W. and Sulem, C., Numerical Simulation of Gravity Waves, J. Comput. Phys., 1993, vol. 108, no. 1, pp. 73–83.

    MathSciNet  MATH  Google Scholar 

  20. Cuccagna, S., The Hamiltonian Structure of the Nonlinear Schrödinger Equation and the Asymptotic Stability of Its Ground States, Comm. Math. Phys., 2011, vol. 305, no. 2, pp. 279–331.

    MathSciNet  MATH  Google Scholar 

  21. Delort, J.-M., Quasi-Linear Perturbations of Hamiltonian Klein – Gordon Equations on Spheres, Mem. Amer. Math. Soc., vol. 234, no. 1103, Providence, R.I.: AMS, 2015.

    MATH  Google Scholar 

  22. Deng, Y., Ionescu, A. D., Pausader, B., and Pusateri, F., Global Solutions for the 3D Gravity-Capillary Water Waves System, Acta Math., 2017, vol. 219, no. 2, pp. 213–402.

    MathSciNet  MATH  Google Scholar 

  23. Deng, Y., Ionescu, A. D., and Pusateri, F., On the Wave Turbulence Theory of 2D Gravity Waves: 1. Deterministic Energy Estimates, arXiv:2211.10826 (2022).

  24. Germain, P., Masmoudi, N., and Shatah, J., Global Solutions for the Gravity Water Waves Equation in Dimension \(3\), Ann. of Math. (2), 2012, vol. 175, no. 2, pp. 691–754.

    MathSciNet  MATH  Google Scholar 

  25. Harrop-Griffiths, B., Ifrim, M., and Tataru, D., Finite Depth Gravity Water Waves in Holomorphic Coordinates, Ann. PDE, 2017, vol. 3, no. 1, Paper No. 4, 102 pp.

    MathSciNet  MATH  Google Scholar 

  26. Hunter, J. K., Ifrim, M., and Tataru, D., Two Dimensional Water Waves in Holomorphic Coordinates, Comm. Math. Phys., 2016, vol. 346, no. 2, pp. 483–552.

    MathSciNet  MATH  Google Scholar 

  27. Ifrim, M. and Tataru, D., Two Dimensional Water Waves in Holomorphic Coordinates: 2. Global Solutions, Bull. Soc. Math. France, 2016, vol. 144, no. 2, pp. 369–394.

    MathSciNet  MATH  Google Scholar 

  28. Ifrim, M. and Tataru, D., The Lifespan of Small Data Solutions in Two Dimensional Capillary Water Waves, Arch. Ration. Mech. Anal., 2017, vol. 225, no. 3, pp. 1279–1346.

    MathSciNet  MATH  Google Scholar 

  29. Ifrim, M. and Tataru, D., Two-Dimensional Gravity Water Waves with Constant Vorticity: 1. Cubic Lifespan, Anal. PDE, 2019, vol. 12, no. 4, pp. 903–967.

    MathSciNet  MATH  Google Scholar 

  30. Ionescu, A. D. and Pusateri, F., Global Solutions for the Gravity Water Waves System in 2D, Invent. Math., 2015, vol. 199, no. 3, pp. 653–804.

    MathSciNet  MATH  Google Scholar 

  31. Ionescu, A. and Pusateri, F., Global Regularity for 2D Water Waves with Surface Tension, Mem. Amer. Math. Soc., vol. 256, no. 1227, Providence, R.I.: AMS, 2018.

    MATH  Google Scholar 

  32. Ionescu, A. and Pusateri, F., Long-Time Existence for Multi-Dimensional Periodic Water Waves, Geom. Funct. Anal., 2019, vol. 29, no. 3, pp. 811–870.

    MathSciNet  MATH  Google Scholar 

  33. Lannes, D., Well-Posedness of the Water-Waves Equations, J. Amer. Math. Soc., 2005, vol. 18, no. 3, pp. 605–654.

    MathSciNet  MATH  Google Scholar 

  34. Nalimov, V. I., The Cauchy – Poisson Problem, Dinam. Sploshn. Sredy, 1974, vol. 18, pp. 104–210, 254 (Russian).

    MathSciNet  Google Scholar 

  35. Schweizer, B., On the Three-Dimensional Euler Equations with a Free Boundary Subject to Surface Tension, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2005, vol. 22, no. 6, pp. 753–781.

    MathSciNet  MATH  Google Scholar 

  36. Sideris, Th. C., Formation of Singularities in Three-Dimensional Compressible Fluids, Comm. Math. Phys., 1985, vol. 101, no. 4, pp. 475–485.

    MathSciNet  MATH  Google Scholar 

  37. Wahlén, E., A Hamiltonian Formulation of Water Waves with Constant Vorticity, Lett. Math. Phys., 2007, vol. 79, no. 3, pp. 303–315.

    MathSciNet  MATH  Google Scholar 

  38. Wu, S., Well-Posedness in Sobolev Spaces of the Full Water Wave Problem in \(2\)-D, Invent. Math., 1997, vol. 130, no. 1, pp. 39–72.

    MathSciNet  MATH  Google Scholar 

  39. Wu, S., Well-Posedness in Sobolev Spaces of the Full Water Wave Problem in \(3\)-D, J. Amer. Math. Soc., 1999, vol. 12, no. 2, pp. 445–495.

    MathSciNet  MATH  Google Scholar 

  40. Wu, S., Almost Global Wellposedness of the \(2\)-D Full Water Wave Problem, Invent. Math., 2009, vol. 177, no. 1, pp. 45–135.

    MathSciNet  MATH  Google Scholar 

  41. Wu, S., Global Wellposedness of the \(3\)-D Full Water Wave Problem, Invent. Math., 2011, vol. 184, no. 1, pp. 125–220.

    MathSciNet  MATH  Google Scholar 

  42. Wu, S., The Quartic Integrability and Long Time Existence of Steep Water Waves in 2D, arXiv:2010.09117 (2020).

  43. Yosihara, H., Gravity Waves on the Free Surface of an Incompressible Perfect Fluid of Finite Depth, Publ. Res. Inst. Math. Sci., 1982, vol. 18, no. 1, pp. 49–96.

    MathSciNet  MATH  Google Scholar 

  44. Zakharov, V. E., Stability of Periodic Waves of Finite Amplitude on the Surface of a Deep Fluid, J. Appl. Mech. Tech. Phys., 1968, vol. 9, no. 2, pp. 190–194; see also: Prikl. Mekh. Tekhn. Fiz., 1968, vol. 9, no. 2, pp. 86-94.

    Google Scholar 

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Funding

Research supported by PRIN 2020 (2020XB3EFL001) “Hamiltonian and dispersive PDEs”.

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Authors and Affiliations

  1. SISSA, Via Bonomea 265, 34136, Trieste, Italy

    Massimiliano Berti, Alberto Maspero & Federico Murgante

  2. Universita degli studi di Trieste, Dipartimento di Matematica e Geoscienze, Via Valerio 12/1, 34127, Trieste, Italy

    Federico Murgante

Authors
  1. Massimiliano Berti
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  2. Alberto Maspero
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  3. Federico Murgante
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Corresponding authors

Correspondence to Massimiliano Berti, Alberto Maspero or Federico Murgante.

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The authors declare that they have no conflicts of interest.

Additional information

Dedicated to Alain Chenciner on the occasion of his 80th birthday

MSC2010

76B15, 37K55, 37J40

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Berti, M., Maspero, A. & Murgante, F. Hamiltonian Paradifferential Birkhoff Normal Form for Water Waves. Regul. Chaot. Dyn. 28, 543–560 (2023). https://doi.org/10.1134/S1560354723040032

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  • DOI: https://doi.org/10.1134/S1560354723040032

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