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

Non-existence of global energy minimisers in Stokes waves problems

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.
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., http://arxiv.org/abs/1207.7198. 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., http://arxiv.org/abs/1207.7198. 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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