Comptes Rendus
no. 7-8
Partial Differential Equations/Mathematical Problems in Mechanics
On the existence and conditional energetic stability of solitary water waves with weak surface tension
[Sur l'existence et la stabilité des ondes solitaires en présence d'une faible tension superficielle]

Présenté par : Gérard Iooss

Mark D. Groves1, 2 ; E. Wahlén1, 3
1 Fachrichtung 6.1 – Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
2 Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK
3 Department of Mathematics, Lund University, 22100 Lund, Sweden
Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 397-402.

Récemment, Buffoni (2005, 2009) [2,3] a développé une théorie d'existence et de stabilité des ondes solitaires de surface dans le cas d'une faible tension de surface. Cette théorie, qui est de nature variationnelle, repose sur l'hypothèse que l'infimum d'une certaine fonctionnelle variationnelle est strictement sous-homogène par rapport à un petit paramètre. Dans cette Note, on démontre cette propriété de sous-homogénéité stricte complétant ansi la théorie de Buffoni.

An existence and stability theory for solitary water waves with weak surface tension has recently been given by Buffoni (2005, 2009) [2,3]. The theory, which is variational in nature, relies upon the assumption that the infimum of the variational functional is strictly subhomogeneous with respect to a small parameter. In this Note we rigorously establish the relevant strict-subhomogeneity property and thus complete Buffoni's theory.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.01.003
Mark D. Groves 1, 2 ; E. Wahlén 1, 3

1 Fachrichtung 6.1 – Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saarbrücken, Germany
2 Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK
3 Department of Mathematics, Lund University, 22100 Lund, Sweden 
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Mark D. Groves; E. Wahlén. On the existence and conditional energetic stability of solitary water waves with weak surface tension. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 397-402. doi : 10.1016/j.crma.2010.01.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.003/

[1] B. Buffoni Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Ration. Mech. Anal., Volume 173 (2004), pp. 25-68

[2] B. Buffoni Conditional energetic stability of gravity solitary waves in the presence of weak surface tension, Topol. Methods Nonlinear Anal., Volume 25 (2005), pp. 41-68

[3] B. Buffoni Gravity solitary waves by minimization: an uncountable family, Topol. Methods Nonlinear Anal., Volume 34 (2009), pp. 339-352

[4] B. Buffoni; M.D. Groves A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory, Arch. Ration. Mech. Anal., Volume 146 (1999), pp. 183-220

[5] G. Iooss; K. Kirchgässner Bifurcation d'ondes solitaires en présence d'une faible tension superficielle, C. R. Acad. Sci. Paris, Sér. I, Volume 311 (1990), pp. 265-268

[6] M.D. Groves, E. Wahlén, On the existence and conditional energetic stability of solitary gravity-capillary surface waves on deep water, 2010, preprint

[7] M.D. Groves, E. Wahlén, Existence and conditional energetic stability of solitary gravity-capillary water waves with constant vorticity, 2010, preprint

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