Acta Mathematica

Global bifurcation of steady gravity water waves with critical layers

Adrian Constantin, Walter Strauss, and Eugen Vărvărucă

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Abstract

We construct families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed, in particular establishing the existence of waves of large amplitude. A Riemann–Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudodifferential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler–Lagrange equation of an energy functional, enable us to develop a theory of analytic global bifurcation.

Article information

Source
Acta Math., Volume 217, Number 2 (2016), 195-262.

Dates
Received: 30 June 2014
Revised: 26 February 2016
First available in Project Euclid: 17 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1502989201

Digital Object Identifier
doi:10.1007/s11511-017-0144-x

Mathematical Reviews number (MathSciNet)
MR3689941

Zentralblatt MATH identifier
1375.35294

Rights
2016 © Institut Mittag-Leffler

Citation

Constantin, Adrian; Strauss, Walter; Vărvărucă, Eugen. Global bifurcation of steady gravity water waves with critical layers. Acta Math. 217 (2016), no. 2, 195--262. doi:10.1007/s11511-017-0144-x. https://projecteuclid.org/euclid.acta/1502989201


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References

  • Amick C.J., Toland J.F.: On periodic water-waves and their convergence to solitary waves in the long-wave limit. Philos. Trans. Roy. Soc. A 303, 633–669 (1981)
  • Babenko K. I., Some remarks on the theory of surface waves of finite amplitude. Dokl. Akad. Nauk SSSR, 294 (1987), 1033–1037 (Russian); English translation in Soviet Math. Dokl., 35 (1987), 599–603.
  • Buffoni B., Dancer E.N., Toland J.F.: The regularity and local bifurcation of steady periodic water waves. Arch. Ration. Mech. Anal. 152, 207–240 (2000)
  • Buffoni B., Dancer E.N., Toland J.F.: The sub-harmonic bifurcation of Stokes waves. Arch. Ration. Mech. Anal., 152, 241–271 (2000)
  • Buffoni B., Séré É., Toland J.F.: Surface water waves as saddle points of the energy. Calc. Var. Partial Differential Equations, 17, 199–220 (2003)
  • Buffoni B., Toland J.F.: Analytic Theory of Global Bifurcation. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ (2003)
  • Burckel, R. B., An Introduction to Classical Complex Analysis. Vol. 1. Pure and Applied Mathematics, 82. Academic Press, New York–London, 1979.
  • Burton G.R., Toland J.F.: Surface waves on steady perfect-fluid flows with vorticity. Comm. Pure Appl. Math., 64, 975–1007 (2011)
  • Constantin A.: The trajectories of particles in Stokes waves. Invent. Math., 166, 523–535 (2006)
  • Constantin, A., Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, 81. SIAM, Philadelphia, PA, 2011.
  • Constantin A., Ehrnström M., Wahlén E.: Symmetry of steady periodic gravity water waves with vorticity. Duke Math. J., 140, 591–603 (2007)
  • Constantin A., Escher J.: Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech., 498, 171–181 (2004)
  • Constantin A., Escher J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. of Math. 173, 559–568 (2011)
  • Constantin A., Sattinger D., Strauss W.: Variational formulations for steady water waves with vorticity. J. Fluid Mech., 548, 151–163 (2006)
  • Constantin A., Strauss W.: Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math., 57, 481–527 (2004)
  • Constantin A., Strauss W.: Pressure beneath a Stokes wave. Comm. Pure Appl. Math. 63, 533–557 (2010)
  • Constantin A., Strauss W.: Periodic traveling gravity water waves with discontinuous vorticity. Arch. Ration. Mech. Anal., 202, 133–175 (2011)
  • Constantin A., Vărvărucă E.: Steady periodic water waves with constant vorticity: regularity and local bifurcation. Arch. Ration. Mech. Anal. 199, 33–67 (2011)
  • Crandall M.G., Rabinowitz P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal., 8, 321–340 (1971)
  • Dancer E.N.: Bifurcation theory for analytic operators. Proc. London Math. Soc., 26, 359–384 (1973)
  • Dubreil-Jacotin M.-L.: Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pures Appl., 13, 217–291 (1934)
  • Dym, H. McKean, H. P., Fourier Series and Integrals. Probability and Mathematical Statistics, 14. Academic Press, New York–London, 1972.
  • Friedrichs K.: Über ein Minimumproblem für Potentialströmungen mit freiem Rande. Math. Ann. 109, 60–82 (1934)
  • Giaquinta, M. Hildebrandt, S., Calculus of Variations. I. Grundlehren der Mathematischen Wissenschaften, 310. Springer, Berlin–Heidelberg, 1996.
  • Jonsson, I. G., Wave-current interactions, in The Sea (Le Méhauté, B. and Hanes, D.M. eds.), Ocean Eng. Sc., 9(A), pp. 65–120. Wiley, 1990.
  • Ko J., Strauss W.: Effect of vorticity on steady water waves. J. Fluid Mech., 608, 197–215 (2008)
  • Ko J., Strauss W.: Large-amplitude steady rotational water waves. Eur. J. Mech. B Fluids, 27, 96–109 (2008)
  • Luke J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech., 27, 395–397 (1967)
  • Okamoto, H. & Shōji, M., The Mathematical Theory of Permanent Progressive Water-Waves. Advanced Series in Nonlinear Dynamics, 20. World Scientific, River Edge, NJ, 2001.
  • Rabinowitz P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal., 7, 487–513 (1971)
  • Shargorodsky E., Toland J.F.: Bernoulli free-boundary problems. Mem. Amer. Math. Soc., 196, 914 (2008)
  • Spielvogel E.R.: A variational principle for waves of infinite depth. Arch. Ration. Mech. Anal., 39, 189–205 (1970)
  • Stein, E. M.,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993.
  • Strauss W.: Steady water waves. Bull. Amer. Math. Soc., 47, 671–694 (2010)
  • Strauss, W., Rotational steady periodic water waves with stagnation points. Talk at SIAM Conference on Analysis of Partial Differential Equations (San Diego, CA, 2011).
  • Teles da Silva A.F., Peregrine D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech., 195, 281–302 (1988)
  • Thomson, W. (Lord Kelvin), On a disturbing infinity in Lord Rayleigh’s solution for waves in a plane vortex stratum. Nature, 23 (1880), 45–46.
  • Toland J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)
  • Toland J.F.: On a pseudo-differential equation for Stokes waves. Arch. Ration. Mech. Anal., 162, 179–189 (2002)
  • Torchinsky A.: Real-Variable Methods in Harmonic Analysis. Dover, Mineola, NY (2004)
  • Vanden-Broeck J.-M.: Some new gravity waves in water of finite depth. Phys. Fluids, 26, 2385–2387 (1983)
  • Vărvărucă E.: Singularities of Bernoulli free boundaries. Comm. Partial Differential Equations, 31, 1451–1477 (2006)
  • Vărvărucă E.: Some geometric and analytic properties of solutions of Bernoulli free-boundary problems. Interfaces Free Bound., 9, 367–381 (2007)
  • Vărvărucă E.: Bernoulli free-boundary problems in strip-like domains and a property of permanent waves on water of finite depth. Proc. Roy. Soc. Edinburgh Sect. A, 138, 1345–1362 (2008)
  • Vărvărucă E.: On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations, 246, 4043–4076 (2009)
  • Vărvărucă E., Weiss G.S.: A geometric approach to generalized Stokes conjectures. Acta Math., 206, 363–403 (2011)
  • Vărvărucă E., Weiss G.S.: The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincaré Anal. Non Linéaire, 29, 861–885 (2012)
  • Vărvărucă E., Zarnescu A.: Equivalence of weak formulations of the steady water waves equations. Philos. Trans. Roy. Soc. A, 370, 1703–1719 (2012)
  • Wahlén E.: Steady water waves with a critical layer. J. Differential Equations, 246, 2468–2483 (2009)
  • Zakharov, V. E., Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. i Tekh. Fiz., 9 (1968), 86–94 (Russian); English translation in J. Appl. Mech. Tech. Phys., 9 (1968), 190–194.