Steady water waves

Bulletin of the American Mathematical Society

Volumn 47, Issue 4, 2010, Pages 671-694

  Strauss, Walter A ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77957949833     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-2010-01302-1     Document Type: Article

References (91)

1. G. B. Airy, Tides and waves, Encyclopedia Metropolitana 5 (1845), 241-396.
2. B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math. 171 (2008), 485-541.
3. D. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal. 35 (2003), 211-244.
4. D. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci. 5 (2007), 391-430.
5. C. Amick and J. F. Toland, On solitary water-waves of finite amplitude, Arch. Rational Mech. Anal. 76 (1981), 9-95.
6. C. Amick and R. Turner, Small internal waves in two-fluid systems, Arch. Rational Mech. Anal. 108 (1989), 111-139.
7. C. Amick and R. Turner, Center manifolds in equations from hydrodynamics, Nonlinear Diff. Eq. Appl. 1 (1994), 47-90.
8. J. T. Beale, The existence of solitary water waves, Comm. Pure Appl. Math. 5 (1977), 237-275.
9. T. B. Benjamin and J. E. Feir, The disintegration of wavetrains in deep water, J. Fluid Mech. 27 (1967), 417-430.
10. J. Bona, D. Bose, and R. Turner, Finite-amplitude steady waves in stratified fluids, J. Math. Pures Appl. 62 (1983), 389-439.
11. J. Bona, T. Colin, and D. Lannes, Long wave approximations for water waves, Arch. Rat. Mech. Anal. 178 (2005), 373-410.
12. M. J. Boussinesq, Essai sur la théorie des eaux courantes, Mémoirs présentés par divers savants á l'Acad. des Sciences Inst. France (série 2) 23 (1877), 1-680.
13. T. Bridges and A. Mielke, A proof of the Benjamin-Feir instability, Arch. Rat. Mech. Anal. 133 (1995), 145-198.
14. B. Buffoni and J. Toland, Analytic theory of global bifurcation, Princeton, 2003.
15. A. Cauchy, Théorie de la propagation des ondes á la surface d'un fluide pésant, Oeuvres Complètes 1 (1827).
16. A. Constantin, On the deep water wave motion, J. Phys. A 34 (2001), 1405-1417.
17. A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), 523-535.
18. A. Constantin, M. Ehrnström, and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), 591-603.
19. A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), 165-186.
20. A. Constantin, D. Sattinger, and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid Mech. 548 (2006), 151-168.
21. A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity (preprint 2010).
22. A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), no. 4, 481-527.
23. A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Phil. Trans. Roy. Soc 365 (2007), 2227-2239.
24. A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. 60 (2007), 911-950.
25. A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 63 (2010), 533-557.
26. D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc. 20 (2007), 829-930.
27. W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. PDE 10 (1985), 787-1003.
28. W. Craig and D. Nicholls, Traveling gravity water waves in two and three dimensions, Eur. J. Mech. B Fluids 21 (2002), 615-641.
29. W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. P.D.E. 13 (1988), 603-633.
30. W. Craig and C. Sulem, Numerical simulation of gravity waves, J. Comp. Phys. 108 (1993), 73-83.
31. W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: A rigorous approach, Nonlinearity 5 (1992), 497-522.
32. W. Craig and C. E. Wayne, Mathematical aspects of surface waves on water, Russian Math. Surveys 62 (2007), 453-473.
33. A. Craik, The origins of water wave theory, Annu. Rev. Fluid Mech. 36 (2004), 1-28.
34. M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340.
35. A. Teles da Silva and D. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech. 195 (1988), 281-302.
36. A. Dalmedico, La propagation de ondes en eau profonde et ses développements mathématiques, The History of Modern Mathematics, D. Rowe and J. McCleary, eds., 2 (1989), 129-168.
37. E. N. Dancer, Global solution branches for positive mappings, Arch. Rat. Mech. Anal. 52 (1973), 181-192.
38. O. Darrigol, The spirited horse, the engineer, and the mathematician: Water waves in nineteenth-century hydrodynamics, Arch. Hist. Exact Sci. 58 (2003), 21-95.
39. O. Darrigol, Worlds of flow, Oxford, 2005.
40. F. Dias and G. Iooss, Water-waves as a spatial dynamical system, Handbook of mathematical fluid dynamics II, North-Holland, Amsterdam, 2003, pp. 443-499.
41. M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl. 13 (1934), 217-291.
42. L. Euler, General principles of the motion of fluids, Hist. de l'Acad. de Berlin 11 (1755), 274-315.
43. C. Fefferman, Existence and smoothness of the Navier-Stokes equation. The millennium prize problems, Clay Math. Inst., Cambridge, MA, 2006, pp. 57-67.
44. L. E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems, Cambridge, 2000.
45. K. O. Friedrichs, Uber ein minimumproblem für potential strömung mit freien rand, Math. Ann. 109 (1933), 60-82.
46. P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, C. R. Math. Acad. Sci. Paris 347 (2009), 897-902 (and preprint 2009).
47. F. Gerstner, Theorie der wellen, Abhand. Koen. Boehmischen Gesel. Wiss., Prague, 1802.
48. M. Groves, Steady water waves, J. Nonl. Math. Phys. 11 (2004), 435-460.
49. M. Groves and E. Wahlén, Small-amplitude Stokes and solitary waves with an arbitrary distribution of vorticity, Phys. D 237 (2008), 1530-1538.
50. T. Healey and H. Simpson, Global continuation in nonlinear elasticity, Arch. Rat. Mech. Anal. 143 (1998), 1-28.
51. V. M. Hur, Global bifurcation of deep-water waves, SIAM J. Math. Anal. 37 (2006), 1482-1521.
52. V. M. Hur, Exact solitary water waves with vorticity, Arch. Rational Mech. Anal. 188 (2008), 213-244.
53. V. M. Hur, Stokes waves with vorticity (preprint 2009).
54. G. Iooss and P. Plotnikov, Small divisor problem in the theory of three-dimensional water gravity waves, Memoirs Amer. Math. Soc. (2009).
55. R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge University Press, 1997.
56. T. Kano and T. Nishida, Sur les ondes de surface de l'eau avec une justification mathématique deséquations des ondes en eau peu profonde, J. Math. Kyoto Univ. 19 (1979), 335-370.
57. G. Keady and J. Norbury, On the existence theory for irrotational water waves, Math. Proc. Cambridge Phil. Soc. 83 (1978), 137-157.
58. S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl.Math. 38 (1985), 631-641.
59. Joy Ko and W. Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids 27 (2007), 96-109.
60. Joy Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech. 608 (2008), 197-215.
61. D. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, Phil. Mag. 39 (1895), 422-443.
62. Yu. Krasovskii, On the theory of steady-state waves of finite amplitude, USSR Comput. Math. and Math. Phys. 1 (1961), 996-1018.
63. D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc. 18, 605-654.
64. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248.
65. T. Levi-Civita, Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda, Rend. Accad. Lincei 33 (1924), 141-150.
66. G. Lieberman and N. Trudinger, Nonlinear oblique boundary value problems for nonlinear elliptic equations, Trans. Amer. Math. Soc. 295 (1986), 509-546.
67. Zhiwu Lin, On linear instability of 2D solitary water waves, Inter. Math. Res. Not. 2009, no. 7, 1247-1303.
68. H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys. 260 (2005), 319-392.
69. H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2) 162 (2005), 109-194.
70. A. Nekrasov, On steady waves, Izv. Ivanovo-Voznesenk. Politekhn. 3 (1921).
71. H. Okamoto and M. Shoji, The mathematical theory of permanent progressive water-waves, Advanced Series in Nonlinear Dynamics, vol. 20, World Scientific Publishing Co. Inc., River Edge, NJ, 2001.
72. R. Pego and M. Weinstein, Convective linear stability of solitary waves for Boussinesq equations, Stud. in Appl. Math. 99 (1997), 311-375.
73. P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513.
74. G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem: The case of zero surface tension, Comm. Pure Appl. Math. 53 (2000), 1475-1535.
75. B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 753-781.
76. J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685-696.
77. J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math. 61 (2008), 698-744.
78. G. Stokes, On the theory of oscillatory waves, Trans. Camb. Phil. Soc. 8 (1847), 441-455.
79. J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1-48.
80. E. Varvaruca, On some properties of travelling water waves with vorticity, SIAM J. Math. Anal. 39 (2008), 1686-1692.
81. E. Varvaruca, On the existence of extreme waves and the Stokes conjecture with vorticity, J. Diff. Eq. 246 (2009), 4043-4076.
82. E.Wahlén, A note on steady gravity waves with vorticity, Int. Math. Res.Not. 2005, 389-396.
83. E.Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J.Math. Anal. 38 (2006), 921-943.
84. S. Walsh, Steady periodic gravity waves with surface tension (preprint 2010).
85. S. Walsh, Stratified steady periodic water waves, SIAM J. Math. Anal. 41, (2009), 1054-1105.
86. Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math. 130 (1997), 39-72.
87. Sijue Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc. 12 (1999), 445-495.
88. Sijue Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math. 177 (2009), 45-135.
89. Sijue Wu, Global wellposedness of the 3-D full water wave problem (preprint 2009).
90. V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 2 (1968), 190-194.
91. P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math. 61 (2008), 877-940.