Minimization methods for quasi-linear problems with an application to periodic water waves

SIAM Journal on Mathematical Analysis

Volumn 36, Issue 4, 2005, Pages 1080-1094

  Buffoni, B ^a   Sere E ^b   Toland, J F ^c  

^a EPFL   (Switzerland)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

^c University of Bath   (United Kingdom)

Author keywords

Critical point theory; Free boundaries; Minimization; Periodic water waves; Quasi linear elliptic problems; Variational method

Indexed keywords

BIFURCATION (MATHEMATICS); BOUNDARY CONDITIONS; CAPILLARITY; LINEAR EQUATIONS; MATHEMATICAL MODELS; WATER WAVES;

CRITICAL-POINT THEORY; FREE BOUNDARIES; PERIODIC WATER WAVES; QUASI-LINEAR ELLIPTIC PROBLEMS; VARIATIONAL METHOD;

PROBLEM SOLVING;

EID: 22544450503     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141003432766     Document Type: Article

References (24)

1. K. I. BABENKO, On a local existence theorem in the theory of surface waves of finite amplitude, Soviet Math. Dokl., 35 (1987), pp. 647-650.
2. K. I. BABENKO, Some remarks on the theory of surface waves of finite amplitude, Soviet Math. Dokl., 35 (1987), pp. 599-603.
3. K. I. BABENKO, V. YU. PETROVICH, AND A. I. RAKHMANOV, A computational experiment in the theory of surface waves of finite amplitude, Soviet Math. Dokl., 38 (1989), pp. 327-331.
4. K. I. BABENKO, V. YU. PETROVICH, AND A. I. RAKHMANOV, On a demonstrative experiment in the theory of surface waves of finite amplitude, Soviet Math. Dokl., 38 (1989), pp. 626-630.
5. C. BAESENS AND R. S. MACKAY, Uniformly travelling water waves from a dynamical systems viewpoint: Some insight into bifurcations from Stokes' family, J. Fluid Mech., 241 (1992), pp. 333-347.
6. B. BUFFONI, E. N. DANCER, AND J. F. TOLAND, The regularity and local bifurcation of steady periodic water waves, Arch. Ration. Mech. Anal., 152 (2000), pp. 207-240.
7. B. BUFFONI, E. N. DANCER, AND J. F. TOLAND, The sub-harmonic bifurcation of Stokes waves, Arch. Ration. Mech. Anal., 152 (2000), pp. 241-271.
8. B. BUFFONI, É. SÉRÉ, AND J. F. TOLAND, Surface water waves as saddle points of the energy, Calc. Var. Partial Differential Equations, 17 (2003), pp. 199-220.
9. B. CHEN AND P. G. SAFFMAN, Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud. Appl. Math., 62 (1980), pp. 1-21.
10. W. CRAIG AND D. P. NICHOLLS, Travelling two and three dimensional capillary gravity water waves, SIAM J. Math. Anal., 32 (2000), pp. 323-359.
11. G. D. CRAPPER, An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech., 2 (1957), pp. 532-540.
12. F. DIAS AND C. KHARIF, Nonlinear gravity and capillary-gravity waves, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech. 31, Annual Reviews, Palo Alto, CA, 1999, pp. 301-346.
13. J. L. HAMMACK AND D. M. HENDERSON, Resonant interactions among surface water waves, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech. 25, Annual Reviews, Palo Alto, CA, 1993, pp. 55-97.
14. M. C. W. JONES AND J. F. TOLAND, Symmetry and the bifurcation of capillary-gravity waves, Arch. Ration. Mech. Anal., 96 (1986), pp. 29-53.
15. M. A. KRASNOSEL'SKII, Topological Methods for Nonlinear Eigenvalue Problems, Pergamon Press, Oxford, UK, 1964.
16. M. S. LONGUET-HIGGINS, Bifurcation in gravity waves, J. Fluid Mech., 151 (1985), pp. 457-475.
17. J. REEDER AND M. SHINBROT, On Wilton ripples, 1: Formal derivation of the phenomenon, Wave Motion, 3 (1981), pp. 115-135.
18. L. W. SCHWARTZ AND J. D. FENTON, Strongly nonlinear waves, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech. 14, Annual Reviews, Palo Alto, CA, 1982, pp. 39-60.
19. G. G. STOKES, On the theory of oscillatory waves, Trans. Camb. Phil. Soc., 8 (1847), pp. 441455.
20. C. A. STUART, An introduction to bifurcation theory based on differential calculus, Nonlinear Mathematics and Mechanics: Heriot-Watt Symposium, Vol. IV, R. J. Knops, ed., Res. Notes in Math. 39, Pitman, Boston, 1979, pp. 76-132.
21. 1), Ark. Mat., 39 (2001), pp. 383-394.
22. R. E. L. TURNER, A variational approach to surface solitary waves, J. Differential Equations, 55 (1984), pp. 401-438.
23. R. W. YEUNG, Numerical methods in free-surface flows, in Annual Review of Fluid Mechanics, Annu. Rev. Fluid Mech. 14, Annual Reviews, Palo Alto, CA, 1982, pp. 395-442.
24. A. ZYGMUND, Trigonometric Series I & II, Cambridge University Press, Cambridge, UK, 1959.