×

Non-existence of global energy minimisers in Stokes waves problems. (English) Zbl 1292.76024

Summary: 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.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35C07 Traveling wave solutions
49J99 Existence theories in calculus of variations and optimal control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. Baldi, Steady periodic water waves under nonlinear elastic membranes,, J. Reine Angew. Math., 652, 67 (2011) · Zbl 1277.35281 · doi:10.1515/CRELLE.2011.015
[2] B. Buffoni, On the stability of travelling waves with vorticity obtained by minimisation,, to appear in Nonlinear Differential Equations Appl. · Zbl 1277.35283 · doi:10.1007/s00030-013-0223-4
[3] G. R. Burton, Surface waves on steady perfect-fluid flows with vorticity,, Comm. Pure Appl. Math., LXIV, 975 (2011) · Zbl 1277.35284 · doi:10.1002/cpa.20365
[4] A. Constantin, Exact steady periodic water waves with vorticity,, Comm. Pure Appl. Math., LVII, 481 (2004) · Zbl 1038.76011 · doi:10.1002/cpa.3046
[5] A. Constantin, Stability properties of steady water waves with vorticity,, Comm. Pure Appl. Math., LX, 911 (2007) · Zbl 1125.35081 · 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, 217 (1934) · Zbl 0010.22702
[7] V. Kozlov, Steady free-surface vortical flows parallel to the horizontal bottom,, Quart. J. Mech. Appl. Math., 64, 371 (2011) · Zbl 1248.76026 · doi:10.1093/qjmam/hbr010
[8] E. Shargorodsky, <em>Bernoulli Free-Boundary Problems</em>,, Memoirs of Amer. Math. Soc., 0065 (2008) · Zbl 1167.35001 · doi:10.1090/memo/0914
[9] W. A. Strauss, Steady water waves,, Bull. Am. Math. Soc. (N.S.), 47, 671 (2010) · Zbl 1426.76078 · doi:10.1090/S0273-0979-2010-01302-1
[10] J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7, 1 (1996) · Zbl 0897.35067
[11] J. F. Toland, Steady periodic hydroelastic waves,, Arch. Rational Mech. Anal., 189, 325 (2008) · Zbl 1147.76008 · doi:10.1007/s00205-007-0104-2
[12] J. F. Toland, <em>Energy-minimising parallel flows with prescribed vorticity distribution</em>,, to appear in Discrete Continuous Dynam. Systems - A. · Zbl 1292.76023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.