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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| Published in | Journal of Differential Equations Vol. 253; no. 9; pp. 2646 - 2656 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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01.11.2012
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Weiss, Georg S. Zhang, Guanghui |
| Author_xml | – sequence: 1 givenname: Georg S. surname: Weiss fullname: Weiss, Georg S. email: weiss@math.uni-duesseldorf.de – sequence: 2 givenname: Guanghui surname: Zhang fullname: Zhang, Guanghui email: zhang@math.uni-duesseldorf.de |
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| Cites_doi | 10.1007/BF02392728 10.1080/03605300600635012 10.4171/IFB/255 10.1111/1467-9590.01408 10.1002/cpa.3046 10.1016/j.anihpc.2012.05.001 10.1007/s11511-011-0066-y 10.1016/j.aim.2005.07.015 10.1090/S0273-0979-2010-01302-1 10.1017/S0022112005007469 10.1512/iumj.2008.57.3433 10.1215/S0012-7094-07-13624-X 10.1002/cpa.20165 10.1016/j.jde.2008.12.018 10.1088/0951-7715/21/5/012 |
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| Keywords | Second variation Singularity Water wave 76B15 Stagnation point 35R35 76B07 Stokes corner |
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| References_xml | – volume: 246 start-page: 4043 year: 2009 end-page: 4076 ident: br0110 article-title: On the existence of extreme waves and the Stokes conjecture with vorticity publication-title: J. Differential Equations contributor: fullname: Varvaruca – volume: 60 start-page: 911 year: 2007 end-page: 950 ident: br0050 article-title: Stability properties of steady water waves with vorticity publication-title: Comm. Pure Appl. Math. contributor: fullname: Strauss – volume: 47 start-page: 671 year: 2010 end-page: 694 ident: br0020 article-title: Steady water waves publication-title: Bull. Amer. Math. Soc. (N.S.) contributor: fullname: Strauss – volume: 31 start-page: 1451 year: 2006 end-page: 1477 ident: br0090 article-title: Singularities of Bernoulli free boundaries publication-title: Comm. 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Math. J. doi: 10.1512/iumj.2008.57.3433 contributor: fullname: Lindgren – volume: 136 start-page: 321 year: 2007 ident: 10.1016/j.jde.2012.07.005_br0160 article-title: An unstable elliptic free boundary problem arising in solid combustion publication-title: Duke Math. J. doi: 10.1215/S0012-7094-07-13624-X contributor: fullname: Monneau – volume: 60 start-page: 911 year: 2007 ident: 10.1016/j.jde.2012.07.005_br0050 article-title: Stability properties of steady water waves with vorticity publication-title: Comm. Pure Appl. Math. doi: 10.1002/cpa.20165 contributor: fullname: Constantin – volume: 246 start-page: 4043 year: 2009 ident: 10.1016/j.jde.2012.07.005_br0110 article-title: On the existence of extreme waves and the Stokes conjecture with vorticity publication-title: J. Differential Equations doi: 10.1016/j.jde.2008.12.018 contributor: fullname: Varvaruca – volume: 21 start-page: 1141 year: 2008 ident: 10.1016/j.jde.2012.07.005_br0200 article-title: On the streamlines and particle paths of gravitational water waves publication-title: Nonlinearity doi: 10.1088/0951-7715/21/5/012 contributor: fullname: Ehrnström |
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| SubjectTerms | Second variation Singularity Stagnation point Stokes corner Water wave |
| Title | The second variation of the stream function energy of water waves with vorticity |
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