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Uspekhi Mat. Nauk, 2007, Volume 62, Issue 3(375), Pages 95–116 (Mi umn6759)  

This article is cited in 18 scientific papers (total in 18 papers)

Mathematical aspects of surface water waves

W. Craiga, C. E. Wayneb

a McMaster University
b Boston University, Department of Mathematics and Statistics

Abstract: The theory of the motion of a free surface over a body of water is a fascinating subject, with a long history in both applied and pure mathematical research, and with a continuing relevance to the enterprises of mankind having to do with the sea. Despite the recent advances in the field (some of which we will hear about during this Workshop on Mathematical Hydrodynamics at the Steklov Institute), and the current focus of the mathematical community on the topic, many fundamental mathematical questions remain. These have to do with the evolution of surface water waves, their approximation by model equations and by computer simulations, the detailed dynamics of wave interactions, such as would produce rogue waves in an open ocean, and the theory (partially probabilistic) of approximating wave fields over large regions by averaged ‘macroscopic’ quantities which satisfy essentially kinetic equations of motion. In this note we would like to point out open problems and some of the directions of current research in the field. We believe that the introduction of new analytical techniques and novel points of view will play an important rôle in the future development of the area.

DOI: https://doi.org/10.4213/rm6759

Full text: PDF file (2190 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2007, 62:3, 453–473

Bibliographic databases:

UDC: 517.958+531.32
MSC: Primary 76B15; Secondary 35Q53, 35Q55
Received: 24.02.2007

Citation: W. Craig, C. E. Wayne, “Mathematical aspects of surface water waves”, Uspekhi Mat. Nauk, 62:3(375) (2007), 95–116; Russian Math. Surveys, 62:3 (2007), 453–473

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. V. Shamin, “Dynamics of ideal liquid with free surface in conformal variables”, Journal of Mathematical Sciences, 160:5 (2009), 537–678  mathnet  crossref  mathscinet  zmath  elib
    2. Eggers J., Fontelos M.A., “The role of self-similarity in singularities of partial differential equations”, Nonlinearity, 22:1 (2009), R1–R44  crossref  mathscinet  zmath  isi  scopus
    3. Fontelos M.A., de la Hoz F., “Singularities in water waves and the Rayleigh–Taylor problem”, J. Fluid Mech., 651 (2010), 211–239  crossref  mathscinet  zmath  isi  elib  scopus
    4. Р. В. Шамин, “Поверхностные волны на воде минимальной гладкости”, Труды Пятой Международной конференции по дифференциальным и функционально-дифференциальным уравнениям (Москва, 17–24 августа, 2008). Часть 1, СМФН, 35, РУДН, М., 2010, 126–140  mathnet  crossref  mathscinet  zmath  scopus
    5. Strauss W.A., “Steady water waves”, Bull. Amer. Math. Soc. (N.S.), 47:4 (2010), 671–694  crossref  mathscinet  zmath  isi  scopus
    6. Baker G.R., Xie Chao, “Singularities in the complex physical plane for deep water waves”, J. Fluid Mech., 685 (2011), 83–116  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. V. I. Nalimov, “Differential properties of the Dirichlet–Neumann operator”, Siberian Math. J., 54:2 (2013), 271–300  mathnet  crossref  mathscinet  isi
    8. Alazard T., Burq N., Zuily C., “On the Cauchy Problem For Gravity Water Waves”, Invent. Math., 198:1 (2014), 71–163  crossref  mathscinet  zmath  isi  scopus
    9. Alazard T., Delort J.-M., “Global Solutions and Asymptotic Behavior For Two Dimensional Gravity Water Waves”, 48, no. 5, 2015, 1149–1238  mathscinet  zmath  isi
    10. Alazard T., Delort J.-M., “Sobolev Estimates For Two Dimensional Gravity Water Waves”, no. 374, 2015, 1+  mathscinet  isi
    11. V. I. Nalimov, “Unique solvability of the water waves problem in Sobolev spaces”, Siberian Math. J., 57:1 (2016), 97–123  mathnet  crossref  crossref  mathscinet  isi  elib
    12. Wang Chao, Zhang ZhiFei, “Break-down criterion for the water-wave equation”, Sci. China-Math., 60:1 (2017), 21–58  crossref  mathscinet  zmath  isi  scopus
    13. Belykh V.N., “On the Evolution of a Finite Volume of Ideal Incompressible Fluid With a Free Surface”, Dokl. Phys., 62:4 (2017), 213–217  crossref  mathscinet  isi  scopus
    14. de Poyferre T., Quang-Huy Nguyen, “A Paradifferential Reduction For the Gravity-Capillary Waves System At Low Regularity and Applications”, Bull. Soc. Math. Fr., 145:4 (2017), 643–710  crossref  mathscinet  zmath  isi  scopus
    15. V. N. Belykh, “Well-posedness of a nonstationary axisymmetric hydrodynamic problem with free surface”, Siberian Math. J., 58:4 (2017), 564–577  mathnet  crossref  crossref  isi  elib  elib
    16. Ionescu A.D., Pusateri F., “Recent Advances on the Global Regularity For Irrotational Water Waves”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2111 (2018), 20170089  crossref  mathscinet  isi  scopus
    17. Stuhlmeier R., Stiassnie M., “Evolution of Statistically Inhomogeneous Degenerate Water Wave Quartets”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2111 (2018), 20170101  crossref  mathscinet  isi  scopus
    18. Deng Yu., Ionescu A.D., Pausader B., Pusateri F., “Global Solutions of the Gravity-Capillary Water-Wave System in Three Dimensions”, Acta Math., 219:2 (2018), 213–402  crossref  mathscinet  zmath  isi
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