American Institute of Mathematical Sciences

Advanced
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 & Continuous Dynamical Systems - A, 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.  doi: 10.1515/CRELLE.2011.015.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

E. Shargorodsky and J. F. Toland, Bernoulli Free-Boundary Problems,, Memoirs of Amer. Math. Soc., (2008), 0065.  doi: 10.1090/memo/0914.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

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.  doi: 10.1515/CRELLE.2011.015.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

E. Shargorodsky and J. F. Toland, Bernoulli Free-Boundary Problems,, Memoirs of Amer. Math. Soc., (2008), 0065.  doi: 10.1090/memo/0914.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[1]

Walter A. Strauss. Vorticity jumps in steady water waves. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101
[2]

Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191
[3]

Grégoire Allaire, Carlos Conca, Luis Friz, Jaime H. Ortega. On Bloch waves for the Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 1-28. doi: 10.3934/dcdsb.2007.7.1
[4]

Adrian Constantin. Dispersion relations for periodic traveling water waves in flows with discontinuous vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1397-1406. doi: 10.3934/cpaa.2012.11.1397
[5]

Mats Ehrnström. Deep-water waves with vorticity: symmetry and rotational behaviour. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 483-491. doi: 10.3934/dcds.2007.19.483
[6]

Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3109-3123. doi: 10.3934/dcds.2014.34.3109
[7]

Philippe Bonneton, Nicolas Bruneau, Bruno Castelle, Fabien Marche. Large-scale vorticity generation due to dissipating waves in the surf zone. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 729-738. doi: 10.3934/dcdsb.2010.13.729
[8]

André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135
[9]

Grégoire Allaire, Tuhin Ghosh, Muthusamy Vanninathan. Homogenization of stokes system using bloch waves. Networks & Heterogeneous Media, 2017, 12 (4) : 525-550. doi: 10.3934/nhm.2017022
[10]

Delia Ionescu-Kruse. Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1475-1496. doi: 10.3934/cpaa.2012.11.1475
[11]

Silvia Sastre-Gomez. Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2669-2680. doi: 10.3934/dcds.2017114
[12]

Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045
[13]

Denys Dutykh, Delia Ionescu-Kruse. Effects of vorticity on the travelling waves of some shallow water two-component systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5521-5541. doi: 10.3934/dcds.2019225
[14]

Ruy Coimbra Charão, Jáuber Cavalcante Oliveira, Gustavo Alberto Perla Menzala. Energy decay rates of magnetoelastic waves in a bounded conductive medium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 797-821. doi: 10.3934/dcds.2009.25.797
[15]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i
[16]

Tony Lyons. Particle trajectories in extreme Stokes waves over infinite depth. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3095-3107. doi: 10.3934/dcds.2014.34.3095
[17]

Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036
[18]

Peter A. Hästö. On the existance of minimizers of the variable exponent Dirichlet energy integral. Communications on Pure & Applied Analysis, 2006, 5 (3) : 415-422. doi: 10.3934/cpaa.2006.5.415
[19]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044
[20]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences