no. 10
Exact periodic traveling water waves with vorticity
[Ondes d'eau avec tourbillons]
Constantin, Adrian1 ; Strauss, Walter2
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, Tome 335 (2002) no. 10, pp. 797-800.

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 :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02565-7
Constantin, Adrian 1 ; Strauss, Walter 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 
@article{CRMATH_2002__335_10_797_0,
     author = {Constantin, Adrian and Strauss, Walter},
     title = {Exact periodic traveling water waves with vorticity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {797--800},
     publisher = {Elsevier},
     volume = {335},
     number = {10},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02565-7},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02565-7/}
}
TY  - JOUR
AU  - Constantin, Adrian
AU  - Strauss, Walter
TI  - Exact periodic traveling water waves with vorticity
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 797
EP  - 800
VL  - 335
IS  - 10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(02)02565-7/
DO  - 10.1016/S1631-073X(02)02565-7
LA  - en
ID  - CRMATH_2002__335_10_797_0
ER  - 
%0 Journal Article
%A Constantin, Adrian
%A Strauss, Walter
%T Exact periodic traveling water waves with vorticity
%J Comptes Rendus. Mathématique
%D 2002
%P 797-800
%V 335
%N 10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(02)02565-7/
%R 10.1016/S1631-073X(02)02565-7
%G en
%F CRMATH_2002__335_10_797_0
Constantin, Adrian; Strauss, Walter. Exact periodic traveling water waves with vorticity. Comptes Rendus. Mathématique, Tome 335 (2002) no. 10, pp. 797-800. doi : 10.1016/S1631-073X(02)02565-7. http://www.numdam.org/articles/10.1016/S1631-073X(02)02565-7/

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

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

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

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

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

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

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

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

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

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

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

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

[15] Lieberman, G.; Trudinger, N. 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] Nekrasov, A.I. On steady waves, Izv. Ivanovo-Voznesenk. Politekhn, Volume 3 (1921)

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

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

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

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

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

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

Cité par Sources :