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Steady periodic capillary waves with vorticity

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Arkiv för Matematik

Abstract

We prove the existence of steady periodic capillary water waves on flows with arbitrary vorticity distributions. They are symmetric two-dimensional waves whose profiles are monotone between crest and trough.

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References

  1. Constantin, A., On the deep water wave motion, J. Phys. A. 34 (2001), 1405–1417.

    Article  MATH  MathSciNet  Google Scholar 

  2. Constantin, A., Edge waves along a sloping beach, J. Phys. A 34 (2001), 9723–9731.

    Article  MATH  MathSciNet  Google Scholar 

  3. Constantin, A. and Escher, J., Symmetry of steady periodic surface water waves with vorticity, J. Fluid. Mech. 498 (2004), 171–181.

    Article  MATH  MathSciNet  Google Scholar 

  4. Constantin, A. and Escher, J., Symmetry of steady deep-water waves with vorticity, European J. Appl. Math. 15 (2004), 755–768.

    Article  MATH  MathSciNet  Google Scholar 

  5. Constantin, A. and Strauss, W., Exact periodic traveling water waves with vorticity, C. R. Math. Acad. Sci. Paris 335 (2002), 797–800.

    MATH  MathSciNet  Google Scholar 

  6. Constantin, A. and Strauss, W., Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), 481–527.

    Article  MATH  MathSciNet  Google Scholar 

  7. Craig, W. and Nicholls, D. P., Traveling two and three dimensional capillary gravity water waves, SIAM J. Math. Anal. 32 (2000), 323–359.

    Article  MATH  MathSciNet  Google Scholar 

  8. Crandall, M. and Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.

    Article  MATH  MathSciNet  Google Scholar 

  9. Crapper, G. D., An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech. 2 (1957), 532–540.

    Article  MATH  MathSciNet  Google Scholar 

  10. Crapper, G. D., Introduction to Water Waves, Ellis Horwood, Chichester; Halsted Press (Wiley), New York, 1984.

  11. Dubreil-Jacotin, M.-L., Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie, J. Math. Pures Appl. 13 (1934), 217–291.

    MATH  Google Scholar 

  12. Ehrnström, M., Uniqueness of steady symmetric deep-water waves with vorticity, J. Nonlinear Math. Phys. 12 (2005), 27–30.

    Article  MATH  MathSciNet  Google Scholar 

  13. Iohvidov, I. S., Krein, M. G. and Langer, H., Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Mathematical Research 9, Akademie-Verlag, Berlin, 1982.

    MATH  Google Scholar 

  14. Johnson, R. S., A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  15. Kalisch, H., Periodic traveling water waves with isobaric streamlines, J. Nonlinear Math. Phys. 11 (2004), 461–471.

    Article  MATH  MathSciNet  Google Scholar 

  16. Kalisch, H., A uniqueness result for periodic traveling waves in water of finite depth, Nonlinear Anal. 58 (2004), 779–785.

    Article  MATH  MathSciNet  Google Scholar 

  17. Kinnersley, W., Exact large amplitude capillary waves on sheets of fluid, J. Fluid Mech. 77 (1976), 229–241.

    Article  MATH  MathSciNet  Google Scholar 

  18. Lamb, H., Hydrodynamics, Cambridge University Press, London, 1924.

    MATH  Google Scholar 

  19. Lighthill, J., Waves in Fluids, Cambridge University Press, Cambridge–New York, 1978.

    MATH  Google Scholar 

  20. Luo, Y. and Trudinger, N. S., Linear second order elliptic equations with Venttsel boundary conditions, Proc. Roy. Soc. Edinburgh A 118 (1991), 193–207.

    MATH  MathSciNet  Google Scholar 

  21. Swan, C., Cummins, I. and James, R., An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid. Mech. 428 (2001), 273–304.

    Article  MATH  Google Scholar 

  22. Teles da Silva, A. F. and Peregrine, D. H., Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech. 195 (1988), 281–302.

    Article  MathSciNet  Google Scholar 

  23. Thomas, G. and Klopman, G., Wave-current interactions in the nearshore region, in Gravity Waves in Water of Finite Depth, Advances in Fluid Mechanics 10, pp. 215–319, Computational Mechanics Publications, Southhampton, United Kingdom, 1997.

    Google Scholar 

  24. Wahlén, E., Uniqueness for autonomous planar differential equations and the Lagrangian formulation of water flows with vorticity, J. Nonlinear Math. Phys. 11 (2004), 549–555.

    Article  MATH  MathSciNet  Google Scholar 

  25. Wahlén, E., On steady gravity waves with vorticity, Int. Math. Res. Not. 2004:54 (2004), 2881–2896.

    Article  MATH  Google Scholar 

  26. Wahlén, E., A note on steady gravity waves with vorticity, Int. Math. Res. Not. 2005:7 (2005), 389–396.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Lund University, P.O. Box 118, SE-221 00, Lund, Sweden

    Erik Wahlén

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  1. Erik Wahlén
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Correspondence to Erik Wahlén.

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Wahlén, E. Steady periodic capillary waves with vorticity. Ark Mat 44, 367–387 (2006). https://doi.org/10.1007/s11512-006-0024-7

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  • DOI: https://doi.org/10.1007/s11512-006-0024-7

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