Surface water waves as saddle points of the energy

Buffoni, Boris; Séré, Eric; Toland, John

dc.contributor.author	Buffoni, Boris
dc.contributor.author	Séré, Eric HAL ID: 171149
dc.contributor.author	Toland, John
dc.date.accessioned	2011-05-13T08:11:56Z
dc.date.available	2011-05-13T08:11:56Z
dc.date.issued	2003
dc.identifier.uri	https://basepub.dauphine.fr/handle/123456789/6276
dc.language.iso	en	en
dc.subject	periodic water waves	en
dc.subject	free-boundary problem	en
dc.subject	variational method	en
dc.subject	critical-point theory	en
dc.subject	saddle point	en
dc.subject	mountain pass	en
dc.subject	Morse index	en
dc.subject.ddc	515	en
dc.title	Surface water waves as saddle points of the energy	en
dc.type	Article accepté pour publication ou publié
dc.description.abstracten	By applying the mountain-pass lemma to an energy functional, we establish the existence of two-dimensional water waves on the surface of an infinitely deep ocean in a constant gravity field. The formulation used, which is due to K. I. Babenko [3, 4] (and later to others, independently), has as its independent variable an amplitude function which gives the surface elevation. Its nonlinear term is purely quadratic but it is nonlocal because it involves the Hilbert transform. Moreover the energy functional from which it is derived is rather degenerate and offers an important challenge in the calculus of variations. In the present treatment the first step is to truncate the integrand, and then to penalize and regularize it. The mountain-pass lemma gives the existence of critical points of the resulting problem. To check that, in the limit of vanishing regularization, the critical points converge to a non-trivial water wave, we need a priori estimates and information on their Morseindex in the spirit of the work by Amann and Zehnder [1] (see also [14]). The amplitudes of the waves so obtained are compared with those obtained from the bifurcation argument of Babenko, and are found to extend the parameter range where existence is known by analytical methods. We also compare our approach with the minimization-under-constraint method used by R. E. L. Turner [25].	en
dc.relation.isversionofjnlname	Calculus of Variations and Partial Differential Equations
dc.relation.isversionofjnlvol	17	en
dc.relation.isversionofjnlissue	2	en
dc.relation.isversionofjnldate	2003
dc.relation.isversionofjnlpages	199-220	en
dc.relation.isversionofdoi	http://dx.doi.org/10.1007/s00526-002-0166-9	en
dc.description.sponsorshipprivate	oui	en
dc.relation.isversionofjnlpublisher	Springer	en
dc.subject.ddclabel	Analyse	en