The second variation of the stream function energy of water waves with vorticity

We compute the second variation of the stream function energy of two-dimensional steady free surface gravity water waves with vorticity in the stream function formulation. We prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with Stokes corner a...

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Bibliographic Details
Published inJournal of Differential Equations Vol. 253; no. 9; pp. 2646 - 2656
Main Authors Weiss, Georg S., Zhang, Guanghui
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.11.2012
Subjects
Second variation
Singularity
Stagnation point
Stokes corner
Water wave
Second variation
Singularity
Water wave
76B15
Stagnation point
35R35
76B07
Stokes corner
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Summary:We compute the second variation of the stream function energy of two-dimensional steady free surface gravity water waves with vorticity in the stream function formulation. We prove that for nonpositive vorticity the second variation of the stream function energy at extreme waves with Stokes corner asymptotics cannot be nonnegative in any small neighbourhood of a given isolated stagnation point. The particular form of our second variation suggests however the possibility that certain singularities in the case of nonzero vorticity might be constructible as minimizers of the stream function energy.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2012.07.005