Exact periodic traveling water waves with vorticity

Adrian Constantin; Walter Strauss

doi:10.1016/S1631-073X(02)02565-7

Exact periodic traveling water waves with vorticity
[Ondes d'eau avec tourbillons]

Adrian Constantin ¹ ; Walter Strauss ²

¹ Department of Mathematics, Lund University, PO Box 118, 22100 Lund, Sweden
² Department of Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, RI 02912, USA

Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 797-800.

Résumé
Abstract

Pour le problème classique des ondes d'eau sans viscosité sous l'influence de la gravité, décrit par l'équation d'Euler avec une surface libre à fond plat, nous construisons des houles aux tourbillons. Ce sont des ondes symmétriques dont les profils sont monotones entre chaque sommet et creux. Nous employons la théorie de la bifurcation globale pour construire un ensemble connexe de telles solutions. Cet ensemble contient des ondes au profil non-oscillatoire et aussi des ondes qui approchent des flots avec des points de stagnation.

For the classical inviscid water wave problem under the influence of gravity, described by the Euler equation with a free surface over a flat bottom, we construct periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use global bifurcation theory to construct a connected set of such solutions. This set contains flat waves as well as waves that approach flows with stagnation points.

Reçu le : 2002-06-18
Accepté le : 2002-09-13
Publié le : 2002-10-05

DOI : 10.1016/S1631-073X(02)02565-7

Affiliations des auteurs :

Adrian Constantin ¹ ; Walter Strauss ²

¹ Department of Mathematics, Lund University, PO Box 118, 22100 Lund, Sweden
² Department of Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, RI 02912, USA

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Adrian Constantin; Walter Strauss. Exact periodic traveling water waves with vorticity. Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 797-800. doi : 10.1016/S1631-073X(02)02565-7. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02565-7/

Bibliographie
Cité par

[1] C. Amick; L. Fraenkel; J. Toland On the Stokes conjecture for the wave of extreme form, Acta Mathematica, Volume 148 (1982), pp. 193-214

[2] R.E. Baddour; S.W. Song The rotational flow of finite amplitude periodic water waves on shear currents, Applied Ocean Research, Volume 20 (1998), pp. 163-171

[3] A. Constantin On the deep water wave motion, J. Phys. A, Volume 34 (2001), pp. 1405-1417

[4] A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity, 2002 (in preparation)

[5] M. Crandall; P. Rabinowitz Bifurcation from simple eigenvalues, J. Funct. Anal, Volume 8 (1971), pp. 321-340

[6] M.-L. Dubreil-Jacotin Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl, Volume 13 (1934), pp. 217-291

[7] F. Gerstner Theorie der Wellen, Abh. Königl. Böhm. Ges. Wiss, 1802

[8] R. Goyon Contribution à la théorie des houles, Ann. Fac. Sci. Univ. Toulouse, Volume 22 (1958), pp. 1-55

[9] T. Healey; H. Simpson Global continuation in nonlinear elasticity, Arch. Rat. Mech. Anal, Volume 143 (1998), pp. 1-28

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

[11] G. Keady; J. Norbury On the existence theory for irrotational water waves, Math. Proc. Cambridge Philos. Soc, Volume 83 (1978), pp. 137-157

[12] H. Kielhöfer Multiple eigenvalue bifurcation for Fredholm operators, J. Reine Angew. Math, Volume 358 (1985), pp. 104-124

[13] Yu.P. Krasovskii On the theory of steady-state waves of finite amplitude, USSR Comput. Math. Phys, Volume 1 (1961), pp. 996-1018

[14] T. Levi-Civita Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei, Volume 33 (1924), pp. 141-150

[15] G. Lieberman; N. Trudinger Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc, Volume 295 (1986), pp. 509-546

[16] J.B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, Univ. of Wisconsin M.R.C. Report No. 2041, 1979

[17] A.I. Nekrasov On steady waves, Izv. Ivanovo-Voznesenk. Politekhn, Volume 3 (1921)

[18] P. Rabinowitz Some global results for nonlinear eigenvalue problems, J. Funct. Anal, Volume 7 (1971), pp. 487-513

[19] J. Serrin A symmetry property in potential theory, Arch. Rat. Mech. Anal, Volume 43 (1971), pp. 304-318

[20] G. Stokes On the theory of oscillatory waves, Trans. Cambridge Philos. Soc, Volume 8 (1847), pp. 441-455

[21] D. Struik Détermination rigoureuse des ondes irrotationelles périodiques dans un canal á profondeur finie, Math. Ann, Volume 95 (1926), pp. 595-634

[22] J. Toland On the existence of a wave of greatest height and Stokes's conjecture, Proc. Roy. Soc. London A, Volume 363 (1978), pp. 469-485

[23] J. Toland Stokes waves, Topological Meth. Nonl. Anal, Volume 7 (1996), pp. 1-48

[24] E. Zeidler Existenzbeweis für permanente Kapillar/Schwerewellen mit allgemeine Wirbelverteilungen, Arch. Rat. Mech. Anal, Volume 50 (1973), pp. 34-72

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Long Pei On the regularity and symmetry of periodic traveling solutions to weakly dispersive equations with cubic nonlinearity, Mathematical Methods in the Applied Sciences, Volume 46 (2023) no. 6, p. 6403 | DOI:10.1002/mma.8913
Gabriele Bruell; Long Pei Symmetry of Periodic Traveling Waves for Nonlocal Dispersive Equations, SIAM Journal on Mathematical Analysis, Volume 55 (2023) no. 1, p. 486 | DOI:10.1137/21m1433162
Yuqian Zhou; Shanshan Cai; Qian Liu Bounded Traveling Waves of the (2+1)-Dimensional Zoomeron Equation, Mathematical Problems in Engineering, Volume 2015 (2015), p. 1 | DOI:10.1155/2015/163597
YUQIAN ZHOU; QIAN LIU; WEINIAN ZHANG BOUNDED TRAVELING WAVES OF THE GENERALIZED BURGERS–FISHER EQUATION, International Journal of Bifurcation and Chaos, Volume 23 (2013) no. 03, p. 1350054 | DOI:10.1142/s0218127413500545
R. S. Johnson A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques, Communications on Pure and Applied Analysis, Volume 11 (2012) no. 4, p. 1497 | DOI:10.3934/cpaa.2012.11.1497
Hung-Chu Hsu; Chiu-On Ng; Hwung-Hweng Hwung A New Lagrangian Asymptotic Solution for Gravity–Capillary Waves in Water of Finite Depth, Journal of Mathematical Fluid Mechanics, Volume 14 (2012) no. 1, p. 79 | DOI:10.1007/s00021-010-0045-7
Frederic Rousset; Nikolay Tzvetkov Transverse instability of the line solitary water-waves, Inventiones mathematicae, Volume 184 (2011) no. 2, p. 257 | DOI:10.1007/s00222-010-0290-7
Yuqian Zhou; Qian Liu; Weinian Zhang Bounded traveling waves of the Burgers–Huxley equation, Nonlinear Analysis: Theory, Methods Applications, Volume 74 (2011) no. 4, p. 1047 | DOI:10.1016/j.na.2010.09.012
R.S. Johnson Periodic waves over constant vorticity: Some asymptotic results generated by parameter expansions, Wave Motion, Volume 46 (2009) no. 6, p. 339 | DOI:10.1016/j.wavemoti.2009.06.006
Mats Ehrnström; Joachim Escher; Bogdan-Vasile Matioc Two-dimensional steady edge waves. Part I: Periodic waves, Wave Motion, Volume 46 (2009) no. 6, p. 363 | DOI:10.1016/j.wavemoti.2009.06.002
Adrian Constantin; Mats Ehrnström; Gabriele Villari Particle trajectories in linear deep-water waves, Nonlinear Analysis: Real World Applications, Volume 9 (2008) no. 4, p. 1336 | DOI:10.1016/j.nonrwa.2007.03.003
Adrian Constantin; Walter A. Strauss Stability properties of steady water waves with vorticity, Communications on Pure and Applied Mathematics, Volume 60 (2007) no. 6, p. 911 | DOI:10.1002/cpa.20165
Adrian Constantin; Mats Ehrnström; Erik Wahlén Symmetry of steady periodic gravity water waves with vorticity, Duke Mathematical Journal, Volume 140 (2007) no. 3 | DOI:10.1215/s0012-7094-07-14034-1
David Henry Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves, Journal of Nonlinear Mathematical Physics, Volume 14 (2007) no. 1, p. 1 | DOI:10.2991/jnmp.2007.14.1.1
Vera Mikyoung Hur Symmetry of steady periodic water waves with vorticity, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 365 (2007) no. 1858, p. 2203 | DOI:10.1098/rsta.2007.2002
Adrian Constantin; Walter Strauss Rotational steady water waves near stagnation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 365 (2007) no. 1858, p. 2227 | DOI:10.1098/rsta.2007.2004
David Henry Particle trajectories in linear periodic capillary and capillary–gravity water waves, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 365 (2007) no. 1858, p. 2241 | DOI:10.1098/rsta.2007.2005
Erik Wahlén Steady periodic capillary waves with vorticity, Arkiv för Matematik, Volume 44 (2006) no. 2, p. 367 | DOI:10.1007/s11512-006-0024-7
D. Henry The trajectories of particles in deep-water Stokes waves, International Mathematics Research Notices (2006) | DOI:10.1155/imrn/2006/23405
Mats Ehrnström A Note on Surface Profiles for Symmetric Gravity Waves with Vorticity, Journal of Nonlinear Mathematical Physics, Volume 13 (2006) no. 1, p. 1 | DOI:10.2991/jnmp.2006.13.1.1
Mats Ehrnström A unique continuation principle for steady symmetric water waves with vorticity, Journal of Nonlinear Mathematical Physics, Volume 13 (2006) no. 4, p. 484 | DOI:10.2991/jnmp.2006.13.4.3
Mats EHRNSTRÖM Uniqueness of Steady Symmetric Deep-Water Waves with Vorticity, Journal of Nonlinear Mathematical Physics, Volume 12 (2005) no. 1, p. 27 | DOI:10.2991/jnmp.2005.12.1.4
Adrian Constantin; Walter Strauss Exact steady periodic water waves with vorticity, Communications on Pure and Applied Mathematics, Volume 57 (2004) no. 4, p. 481 | DOI:10.1002/cpa.3046
M.D. Groves Steady Water Waves, Journal of Nonlinear Mathematical Physics, Volume 11 (2004) no. 4, p. 435 | DOI:10.2991/jnmp.2004.11.4.2
Henrik Kalisch Periodic Traveling Water Waves with Isobaric Streamlines, Journal of Nonlinear Mathematical Physics, Volume 11 (2004) no. 4, p. 461 | DOI:10.2991/jnmp.2004.11.4.3
Erik Wahlén Uniqueness for Autonomous Planar Differential Equations and the Lagrangian Formulation of Water Flows with Vorticity, Journal of Nonlinear Mathematical Physics, Volume 11 (2004) no. 4, p. 549 | DOI:10.2991/jnmp.2004.11.4.10

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