Journal of Mathematical Sciences

, Volume 170, Issue 3, pp 356–370 | Cite as

Surface waves on the water of minimal smoothness

  • R. V. Shamin


Free Surface Surface Wave Sobolev Space Free Boundary Plasma Physic Report 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    W. Craig and C. Sulem, “Numerical simulation of gravity waves,” J. Comput. Phys., 108, 73–83 (1993).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    W. Craig and C. E. Wayne, “Mathematical aspects of surface water waves,” Russ. Math. Surv., 62, No. 3, 453–472 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    A. I. Dyachenko, E. A. Kuznetsov, M. D. Spector, and V. E. Zakharov, “Analytical description of the free surface dynamics of an ideal liquid (canonical formalism and conformal mapping),” Phys. Lett. A, 221, 73–79 (1996).CrossRefGoogle Scholar
  4. 4.
    A. I. Dyachenko, V. E. Zakharov, and E. A. Kuznetsov, “Nonlinear dynamics of the free surface of an ideal liquid,” Plasma Physics Reports, 22, No. 10, 916–928 (1999).Google Scholar
  5. 5.
    G. H. Hardy, “Weierstrass’s nondifferentiable function,” Trans. Amer. Math. Soc., 17, 301–325 (1916).zbMATHMathSciNetGoogle Scholar
  6. 6.
    V. I. Nalimov, “The Cauchy–Poisson problem,” Dinamika Sploshn. Sredy, 18, 104–210 (1974).MathSciNetGoogle Scholar
  7. 7.
    V. I. Nalimov, “Nonstationary vortex surface waves,” Siberian Math. J., 37, No. 6, 1356–1366 (1996).CrossRefMathSciNetGoogle Scholar
  8. 8.
    V. I. Nalimov and V.V. Pukhnachev, Unsteady Motions of Ideal Liquid with Free Boundary [in Russian], Novosibirsk State University, Novosibirsk (1975).Google Scholar
  9. 9.
    L. Nirenberg, Topics in Nonlinear Functional Analysis [Russian translation], Mir, Moscow (1977).zbMATHGoogle Scholar
  10. 10.
    T. Nishida, “A note on a theorem of Nirenberg,” J. Differential Geom., 12, 629-633 (1977).zbMATHMathSciNetGoogle Scholar
  11. 11.
    L. V. Ovsyannikov, “Justification of the shallow water theory,” Dinamika Sploshn. Sredy, 15, 104–125 (1973).Google Scholar
  12. 12.
    R. V. Shamin, “On a numerical method for problem of a non-stationary flow of incommerssible fluid with a free surface,” Sib. Zh. Vychisl. Mat., 9, No. 4, 379–389 (2006).Google Scholar
  13. 13.
    R. V. Shamin, “On the existence of smooth solutions of Dyachenko equations describing unsteady flows of ideal liquid with a free surface,” Dokl. RAN, 406, No. 5, 112-113 (2006).MathSciNetGoogle Scholar
  14. 14.
    R. V. Shamin, “On an estimate for the existence time of solutions of the Cauchy–Kovalevskaya system with examples from the hydrodynamics of an ideal fluid with a free surface,” J. Math. Sci. (N. Y.), 153, No. 5, 612–628 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. V. Shamin, Numerical Experiments in Simulation of Ocean Surface Waves [in Russian], Nauka, Moscow (2008).Google Scholar
  16. 16.
    R. V. Shamin, “The dynamics of an ideal fluid with a free surface in conformal variables,” Sovrem. Mat. Fundam. Napravl., 28, 3–144 (2008).Google Scholar
  17. 17.
    R. V. Shamin, “On the estimation of the lifetime of solutions of an equation describing surface waves,” Dokl. RAN, 418, No. 5, 112-113 (2008).MathSciNetGoogle Scholar
  18. 18.
    S. Wu, “Well-posedness in Sobolev spaces of the full water wave problem in 2-D,” Invent. Math., 130, 39–72 (1999).CrossRefGoogle Scholar
  19. 19.
    S. Wu, “Well-posedness in Sobolev spaces of the full water wave problem in 3-D,” J. Amer. Math. Soc., 12, No. 2, 445–495 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    V. E. Zakharov, A. I. Dyachenko, and O. A. Vasilyev, “New method for numerical simulation of a nonstationary potential flow of incompressible liquid with a free surface,” Eur. J. Mech. B Fluids, 21, 283–291 (2002).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Shirshov Institute of Oceanology of Russian Academy of SciencesMoscowRussia

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