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Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves

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Abstract

We consider the initial value problem to the Isobe–Kakinuma model for water waves and the structure of the model. The Isobe–Kakinuma model is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian. The Isobe–Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations. Since the hypersurface \(t=0\) is characteristic for the Isobe–Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution. Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces. We also discuss the linear dispersion relation to the model.

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References

  1. Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics. Invent. Math. 171, 485–541 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Baker, G.A., Graves-Morris, P.: Padé Approximants, Encyclopedia of Mathematics and its Applications, vol. 59, 2nd edn. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  3. Boussinesq, J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pure. Appl. 17, 55–108 (1872)

    MathSciNet  MATH  Google Scholar 

  4. Fujiwara, H., Iguchi, T.: A shallow water approximation for water waves over a moving bottom. Adv. Stud. Pure Math. 64, 77–88 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Iguchi, T.: A shallow water approximation for water waves. J. Math. Kyoto Univ. 49, 13–55 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Iguchi, T.: A mathematical analysis of tsunami generation in shallow water due to seabed deformation. Proc. R. Soc. Edinb. Sect. A. 141, 551–608 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Iguchi, T.: Isobe–Kakinuma model for water waves as a higher order shallow water approximation, arXiv:1704.02419

  8. Isobe, M.: A proposal on a nonlinear gentle slope wave equation. In: Proceedings of Coastal Engineering, Japan Society of Civil Engineers, vol. 41, pp. 1–5 (1994) (Japanese)

  9. Isobe, M.: Time-dependent mild-slope equations for random waves. In: Proceedings of 24th International Conference on Coastal Engineering, pp. 285–299. ASCE (1994)

  10. Kakinuma, T.: [title in Japanese], In: Proceedings of Coastal Engineering, Japan Society of Civil Engineers, vol. 47, pp. 1–5 (2000) (Japanese)

  11. Kakinuma, T.: A set of fully nonlinear equations for surface and internal gravity waves. In: Brebbia, C.A. (ed.) Coastal Engineering V: Computer Modelling of Seas and Coastal Regions, pp. 225–234. WIT Press, Southampton (2001)

  12. Kakinuma, T.: A nonlinear numerical model for surface and internal waves shoaling on a permeable beach. In: Brebbia, C.A., Almorza, D., López-Aguayo, F. (eds.) Coastal Engineering VI: Computer Modelling and Experimental Measurements of Seas and Coastal Regions, pp. 227–236. WIT Press, Southampton (2003)

  13. Lannes, D.: Well-posedness of the water-waves equations. J. Am. Math. Soc. 18, 605–654 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lannes, D.: The Water Waves Problem: Mathematical Analysis and Asymptotics, Mathematical Surveys and Monographs, vol. 188. American Mathematical Society, Providence, RI (2013)

    MATH  Google Scholar 

  15. Li, Y.A.: A shallow-water approximation to the full water wave problem. Commun. Pure Appl. Math. 59, 1225–1285 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luke, J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397 (1967)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Murakami, Y., Iguchi, T.: Solvability of the initial value problem to a model system for water waves. Kodai Math. J. 38, 470–491 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39–72 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Wu, S.: Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445–495 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Ryo Nemoto & Tatsuo Iguchi

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  1. Ryo Nemoto
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  2. Tatsuo Iguchi
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Correspondence to Tatsuo Iguchi.

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Communicated by G. P. Galdi

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Nemoto, R., Iguchi, T. Solvability of the Initial Value Problem to the Isobe–Kakinuma Model for Water Waves. J. Math. Fluid Mech. 20, 631–653 (2018). https://doi.org/10.1007/s00021-017-0338-1

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