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Stokeswaves with vorticity

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Abstract

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep body of water under the force of gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite cylinder with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. The proof combines a generalized degree theory, global bifurcation theory, and Whyburn’s lemma in topology with the Schauder theory for elliptic problems and the maximum principle.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Amick, L E. Fraenkel, and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math. 148 (1982), 193–214.

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Amick and J. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), 9–95.

    Article  MATH  MathSciNet  Google Scholar 

  4. K. I. Babenko, Some remarks on the theory of surface waves of finite amplitude, Dokl. Akad. Nauk SSSR 294 (1987), 1289–1292.

    MathSciNet  Google Scholar 

  5. B. Buffoni, E. N. Dancer, and J. F. Toland, The regularity and local bifurcation of steady periodic water waves, Arch. Rational Mech. Anal. 152 (2000), 207–240.

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Buffoni, E. N. Dancer, and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Rational Mech. Anal. 152 (2000), 241–271.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation: An Introduction, Princeton University Press, 2003.

  8. A. Constantin, M. Ehrnström, and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), 591–603.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. (N.S.) 44 (2007), 423–431.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Coddington and N. Levinson, Theory of Ordinary Differential Equations,McGraw-Hill, New York, 1955.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Constantin and W. A. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London, Ser. A 365 (2007), 2227–2239.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Constantin and W. A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. 60 (2007), 911–950.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Constantin and W. A. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 53 (2010), 533–557.

    MathSciNet  Google Scholar 

  17. A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, preprint (2009).

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  20. F. J. von Gerstner, Theorie der wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809), 412–445; also in Abhand. Kön. Böhmischen Gesel. Wiss. (1802).

    Article  Google Scholar 

  21. D. Gilbarg, The Phragmén-Lindelöf theorem for elliptic partial differential equations, J. Rational Mech. Anal. 1 (1952), 411–417.

    MathSciNet  Google Scholar 

  22. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.

    MATH  Google Scholar 

  23. M. Groves and E. Wahlén, Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity, J. Phys. D 237 (2008), 1530–1538.

    Article  MATH  Google Scholar 

  24. T. Healey and H. Simpson, Global continuation in nonlinear elasticity, Arch. Rational Mech. Anal. 143 (1998), 1–28.

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Notes 2006, Art. ID 23405.

  26. V. M. Hur, Global bifurcation of deep-water waves with vorticity, SIAM J. Math. Anal. 37 (2006), 1482–1521.

    Article  MATH  MathSciNet  Google Scholar 

  27. V. M. Hur, Symmetry of steady periodic water waves with vorticity, Philos. Trans. Roy. Soc. London, Ser. A 365 (2007), 2203–2214.

    Article  MATH  MathSciNet  Google Scholar 

  28. V. M. Hur, Exact solitary water waves with vorticity, Arch. Rational Mech. Anal. 188 (2008), 213–244.

    Article  MATH  MathSciNet  Google Scholar 

  29. V. M. Hur, Symmetry of solitary water waves with vorticity, Math. Res. Lett. 15 (2008), 491–509.

    MATH  MathSciNet  Google Scholar 

  30. V. M. Hur and Z. Lin, Unstable surface waves in running water, Comm. Math. Phys. 282 (2008), 73–796.

    Article  MathSciNet  Google Scholar 

  31. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1967.

    Google Scholar 

  32. G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Cambridge Philos. Soc. 83 (1978), 137–157.

    Article  MATH  MathSciNet  Google Scholar 

  33. J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech. 608 (2008), 197–215.

    Article  MATH  MathSciNet  Google Scholar 

  34. J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids 27 (2008), 96–109.

    Article  MATH  MathSciNet  Google Scholar 

  35. Yu. P. Krasov’skii, On the theory of steady waves of finite amplitude, USSR Comput.Math. Math. Phys. 1 (1961), 996–1018.

    Article  MathSciNet  Google Scholar 

  36. N. V. Krylov, Lectures on Elliptic and Parablolic Equations in Hölder Spaces, Amer. Math. Soc., Providence, RI, 1996.

    Google Scholar 

  37. T. Levi-Civita, Détermination rigoureuse des ondes permanentes d’ampleur finie, Math. Ann. 93 (1925), 264–314.

    Article  MATH  MathSciNet  Google Scholar 

  38. G. Lieberman and N. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), 509–546.

    Article  MATH  MathSciNet  Google Scholar 

  39. G. Lieberman and N. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), 509–546.

    Article  MATH  MathSciNet  Google Scholar 

  40. J. Lighthill, Waves in Fluids, Cambridge University Press, 1978.

  41. J. B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, University of Wisconsin MRC Report 2041 (1979).

  42. C. C. Mei, The Applied Dynamics of Ocean Surface Waves, World Scientific, Singapore, 1989.

    MATH  Google Scholar 

  43. A. I. Nekrasov, The exact theory of steady waves on the surface of a heavy fluid, Izdt. Akad. Nauk SSSR, Moscow (1951), Translated as University of Wisconsin MRC Report 813 (1967).

    Google Scholar 

  44. P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.

    Article  MATH  MathSciNet  Google Scholar 

  45. R. Sperb, Maximum Principles and their Applications, Academic Press, New York, 1981.

    MATH  Google Scholar 

  46. G. G. Stokes, Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form, Mathematical and Physical Papers Vol. I (1880), 225–228.

  47. G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, in Gravity Waves in Water of Finite Depth, Adv. Fluid Mech. 10 (1997), 215–319.

  48. J. F. Toland, On the existence of a wave of greatest height and Stokes’s conjecture, Proc. Roy. Soc. London, Ser. A 363 (1978), 469–485.

    Article  MATH  MathSciNet  Google Scholar 

  49. J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), 1–48.

    MATH  MathSciNet  Google Scholar 

  50. E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal. 39 (2008), 1686–1692.

    Article  MATH  MathSciNet  Google Scholar 

  51. E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Differential Equations 246 (2009), 4043–4076.

    Article  MATH  MathSciNet  Google Scholar 

  52. V. Volpert and A. Volpert, Properness and topological degree for general elliptic operators, Abstr. Appl. Anal. 8 (2003), 129–181.

    Article  MathSciNet  Google Scholar 

  53. E. Wahlén, Steady water waves with a critical layer, J. Differential Equations 246 (2009), 2468–2483.

    Article  MATH  MathSciNet  Google Scholar 

  54. G. T. Whyburn, Topological Analysis, Princeton University Press, 1958.

  55. E. Zeidler, Existenzbeweis für permanente Kapillar-Schwerewellen mit allgemeinen Wirbelverteilungen, Arch. Rational Mech. Anal. 50 (1973), 34–72.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Vera Mikyoung Hur

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  1. Vera Mikyoung Hur
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Correspondence to Vera Mikyoung Hur.

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This work was partly supported by NSF grants DMS-0707647 and DMS-1002854.

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Hur, V.M. Stokeswaves with vorticity. JAMA 113, 331–386 (2011). https://doi.org/10.1007/s11854-011-0010-2

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  • DOI: https://doi.org/10.1007/s11854-011-0010-2

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