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August  2014, 34(8): 3211-3217. doi: 10.3934/dcds.2014.34.3211

Non-existence of global energy minimisers in Stokes waves problems

J. F. Toland 1,

1. 

St John's College, Cambridge, CB2 1TP

Received  April 2013 Published  January 2014

Recently it was shown that a wave profile which minimises total energy, elastic plus hydrodynamic, subject to the vorticity distribution being prescribed, gives rise to a steady hydroelastic wave. Using this formulation, the existence of non-trivial minimisers leading to such waves was established for certain non-zero values of the elastic constants used to model the surface. Here we show that when these constants are zero, global minimisers do not exist except in a unique set of circumstances.
Keywords: Stokes waves, Energy-minimizers, waves with vorticity., shape optimisation.
Mathematics Subject Classification: 76B15, 35C07, 49J9.
Citation: J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211
References:
[1]

P. Baldi and J. F. Toland, Steady periodic water waves under nonlinear elastic membranes, J. Reine Angew. Math., 652 (2011), 67-112. doi: 10.1515/CRELLE.2011.015.

[2]

B. Buffoni and G. R. Burton, On the stability of travelling waves with vorticity obtained by minimisation,, to appear in Nonlinear Differential Equations Appl., ().  doi: 10.1007/s00030-013-0223-4.

[3]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., LXIV (2011), 975-1007. doi: 10.1002/cpa.20365.

[4]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., LVII (2004), 481-527. doi: 10.1002/cpa.3046.

[5]

A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., LX (2007), 911-950. doi: 10.1002/cpa.20165.

[6]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291.

[7]

V. Kozlov and N. Kuznetsov, Steady free-surface vortical flows parallel to the horizontal bottom, Quart. J. Mech. Appl. Math., 64 (2011), 371-399. doi: 10.1093/qjmam/hbr010.

[8]

E. Shargorodsky and J. F. Toland, Bernoulli Free-Boundary Problems, Memoirs of Amer. Math. Soc., 914, ISSN 0065-9266, Providence, RI, 2008. doi: 10.1090/memo/0914.

[9]

W. A. Strauss, Steady water waves, Bull. Am. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[10]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-8.

[11]

J. F. Toland, Steady periodic hydroelastic waves, Arch. Rational Mech. Anal., 189 (2008), 325-362. doi: 10.1007/s00205-007-0104-2.

[12]

J. F. Toland, Energy-minimising parallel flows with prescribed vorticity distribution,, to appear in Discrete Continuous Dynam. Systems - A., (). 

show all references

References:
[1]

P. Baldi and J. F. Toland, Steady periodic water waves under nonlinear elastic membranes, J. Reine Angew. Math., 652 (2011), 67-112. doi: 10.1515/CRELLE.2011.015.

[2]

B. Buffoni and G. R. Burton, On the stability of travelling waves with vorticity obtained by minimisation,, to appear in Nonlinear Differential Equations Appl., ().  doi: 10.1007/s00030-013-0223-4.

[3]

G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math., LXIV (2011), 975-1007. doi: 10.1002/cpa.20365.

[4]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., LVII (2004), 481-527. doi: 10.1002/cpa.3046.

[5]

A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math., LX (2007), 911-950. doi: 10.1002/cpa.20165.

[6]

M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291.

[7]

V. Kozlov and N. Kuznetsov, Steady free-surface vortical flows parallel to the horizontal bottom, Quart. J. Mech. Appl. Math., 64 (2011), 371-399. doi: 10.1093/qjmam/hbr010.

[8]

E. Shargorodsky and J. F. Toland, Bernoulli Free-Boundary Problems, Memoirs of Amer. Math. Soc., 914, ISSN 0065-9266, Providence, RI, 2008. doi: 10.1090/memo/0914.

[9]

W. A. Strauss, Steady water waves, Bull. Am. Math. Soc. (N.S.), 47 (2010), 671-694. doi: 10.1090/S0273-0979-2010-01302-1.

[10]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-8.

[11]

J. F. Toland, Steady periodic hydroelastic waves, Arch. Rational Mech. Anal., 189 (2008), 325-362. doi: 10.1007/s00205-007-0104-2.

[12]

J. F. Toland, Energy-minimising parallel flows with prescribed vorticity distribution,, to appear in Discrete Continuous Dynam. Systems - A., (). 

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