Skip to main content
SpringerLink
Log in

Analyticity of the Free Surface for Periodic Travelling Capillary-Gravity Water Waves with Vorticity

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

For capillary-gravity water waves with vorticity, in the absence of stagnation points, we prove that if the vorticity function has a Hölder continuous derivative then the free surface will be smooth. Furthermore, once we assume that the vorticity function is real analytic it will follow that the wave surface profile is also analytic. In particular this scenario includes the case of irrotational fluid flow where the vorticity is zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12, 623–727 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  2. Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Constantin A., Escher J.: Analyticity of periodic travelling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Constantin A., Strauss W.: Pressure and trajectories beneath a Stokes wave. Comm. Pure Appl. Math. 63, 533–557 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Constantin A., Varvaruca E.: Steady periodic water waves with constant vorticity: regularity and local bifurcation. Arch. Rational Mech. Anal. 199, 33–67 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Craig W., Matei A.-M.: On the regularity of the Neumann problem for the free surfaces with surface tension. Proc. Amer. Math. Soc. 135, 2497–2504 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Fraenkel L.E.: An introduction to maximum principles and symmetry in elliptic problems. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  12. Gerstner F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412–445 (1809)

    Article  Google Scholar 

  13. Gilbarg D., Trudinger N.S.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin (2001)

    MATH  Google Scholar 

  14. Henry, D.: On the deep-water Stokes wave flow. Int. Math. Res. Not. Art. ID rnn071, p. 7 (2008)

  15. Henry D.: Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves. J. Nonlinear Math. Phys. 14, 1–7 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Henry D.: Particle trajectories in linear periodic capillary and capillary-gravity water waves. Philos. Trans. R. Soc. Lond. Ser. A 365, 2241–2251 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Henry D.: On Gerstner’s water wave. J. Nonl. Math. Phys. 15, 87–95 (2008)

    Article  MathSciNet  Google Scholar 

  18. Henry D.: Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity. SIAM J. Math. Anal. 42, 3103–3111 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Henry, D.: Regularity for steady periodic capillary water waves with vorticity. preprint

  20. Hsu, H.-C., Ng, C.-O., Hwung, H.-H.: A new Lagrangian asymptotic solution for gravity-capillary waves in water of finite depth. J. Math. Fluid Mech. (to appear). doi:10.1007/s00021-010-0045-7

  21. Johnson R.S.: A modern introduction to the mathematical theory of water waves. Cambridge Univ. Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  22. Kinderlehrer D., Nirenberg L., Spruck J.: Regularity in elliptic free boundary problems. J. Anal. Math. 34, 86–119 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Kinsman B.: Wind waves. Prentice Hall, New Jersey (1965)

    Google Scholar 

  25. Lewy H.: A note on harmonic functions and a hydrodynamical application. Proc. Amer. Math. Soc. 3, 111–113 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lighthill J.: Waves in fluids. Cambridge University Press, Cambridge (1978)

    MATH  Google Scholar 

  27. Matioc, B.V.: Analyticity of the streamlines for periodic travelling water waves with bounded vorticity, IMRN Art. ID rnq235, pp. 14 (2010)

  28. Matioc, B.V.: On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math., to appear

  29. Morrey C.B.: Multiple integrals in the calculus of variations. Springer, Berlin (1966)

    MATH  Google Scholar 

  30. Rankine W.J.M.: On the exact form of waves near the surface of deep water. Phil. Trans. R. Soc. London Ser. A 153, 127–138 (1863)

    Article  Google Scholar 

  31. Wahlén E.: Steady periodic capillary-gravity waves with vorticity. SIAM J. Math. Anal. 38, 921–943 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wahlén E.: Steady water waves with a critical layer. J. Differ. Equ. 246, 2468–2483 (2009)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

    David Henry

Authors
  1. David Henry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Henry.

Additional information

Communicated by A. Constantin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henry, D. Analyticity of the Free Surface for Periodic Travelling Capillary-Gravity Water Waves with Vorticity. J. Math. Fluid Mech. 14, 249–254 (2012). https://doi.org/10.1007/s00021-011-0056-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-011-0056-z

Mathematics Subject Classification (2000)

Keywords

Search

Navigation