no. 5
Mathematical Problems in Mechanics/Partial Differential Equations
Standing waves on infinite depth
[Ondes de gravité stationnaires en profondeur infinie]

Présenté par : Gérard Iooss

Iooss, Gérard1 ; Plotnikov, Pavel2 ; Toland, John3
1 IUF, INLN UMR CNRS-UNSA 6618, 1361, route des Lucioles, 06560 Valbonne, France
2 Lavryentyev Inst. of Hydrodynamics, Siberian Div. Russian Acad. Sci., Lavryentyev pr. 15, Novosibirsk 630090, Russia
3 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Comptes Rendus. Mathématique, Tome 338 (2004) no. 5, pp. 425-431.

On considère le problème des ondes de gravité stationnaires (le clapotis) en profondeur infinie, mis sous la forme d'une EDP du second ordre non locale. Malgré la dégénérescence infinie du problème linéarisé, nous adaptons le théorème des fonctions implicites de Nash–Moser pour montrer l'existence de vagues stationnaires pour un ensemble de valeurs de l'amplitude ε ayant 0 comme point de Lebesgue.

The two-dimensional standing wave problem, for an infinitely deep layer, is considered, based on the formulation of the problem as a second order non local PDE. Despite the presence of infinitely many resonances in the linearized problem, we use the Nash–Moser implicit function theorem to prove the existence of standing waves corresponding to values of the amplitude ε having 0 as a Lebesgue point.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.01.002
Iooss, Gérard 1 ; Plotnikov, Pavel 2 ; Toland, John 3

1 IUF, INLN UMR CNRS-UNSA 6618, 1361, route des Lucioles, 06560 Valbonne, France
2 Lavryentyev Inst. of Hydrodynamics, Siberian Div. Russian Acad. Sci., Lavryentyev pr. 15, Novosibirsk 630090, Russia
3 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK 
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Iooss, Gérard; Plotnikov, Pavel; Toland, John. Standing waves on infinite depth. Comptes Rendus. Mathématique, Tome 338 (2004) no. 5, pp. 425-431. doi : 10.1016/j.crma.2004.01.002. http://www.numdam.org/articles/10.1016/j.crma.2004.01.002/

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