Exact steady periodic water waves with vorticity

Communications on Pure and Applied Mathematics

Volumn 57, Issue 4, 2004, Pages 481-527

  Constantin, Adrian ^a   Strauss, Walter ^b  

^a LUND UNIVERSITY   (Sweden)

^b BROWN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1242319077     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.3046     Document Type: Review

References (51)

1. Agmon, S. On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems. Comm. Pure Appl. Math 15 (1962), 119-147.
2. Amick, C. J.; Fraenkel, L. E.; Toland, J. F. On the Stokes conjecture for the wave of extreme form. Acta Math. 148 (1982), 193-214.
3. Amick, C. J.; Toland, J. F. On periodic water-waves and their convergence to solitary waves in the long-wave limit. Phil. Trans. Roy. Soc. London Ser. A 303 (1981), no. 1481, 633-669.
4. Baddour, R. E.; Song, S. W. The rotational flow of finite amplitude periodic water waves on shear currents. Applied Ocean Research 20 (1998), no. 4, 163-171.
5. Benjamin, T. B. The solitary wave on a stream with an arbitrary distribution of vorticity. J. Fluid Mech. 12 (1962), 97-116.
6. Chang H.-K.; Ho, L.-S. Highly nonlinear interactions between a gravity wave and a current of constant vorticity. J. Chinese Inst. Engrs. 18 (1995), 169-183.
7. Constantin, A. On the deep water wave motion. J. Phys. A 34 (2001), no. 7, 1405-1417.
8. Constantin, A.; Strauss, W. Exact periodic traveling water waves with vorticity. C. R. Math. Acad. Sci. Paris 335 (2002), no. 10, 797-800.
9. Craig, W.; Nicholls, D. P. Traveling two and three-dimensional capillary gravity water waves. SIAM J. Math. Anal. 32 (2000), no. 2, 323-359.
10. Craig, W.; Nicholls, D. P. Traveling gravity water waves in two and three dimensions. Eur. J. Mech. B Fluids 21 (2002), no. 6, 615-641.
11. Crandall, M.; Rabinowitz, P. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971), 321-340.
12. Crapper, G. D. Introduction to water waves. Ellis Horwood Series: Mathematics and Its Applications. Ellis Horwood, Chichester; Halsted Press, New York, 1984.
13. Dias, F.; Kharif, C. Nonlinear gravity and capillary-gravity waves. Annual review of fluid mechanics, 301-346, Annual Review of Fluid Mechanics, 31. Annual Reviews, Palo Alto, Calif., 1999.
14. Dias, F.; Kharif, C. Nonlinear gravity and capillary-gravity waves. Annual review of fluid mechanics, 301-346, Annual Review of Fluid Mechanics, 31. Annual Reviews, Palo Alto, Calif., 1999.
15. Dias, F.; Kharif, C. Nonlinear gravity and capillary-gravity waves. Annual review of fluid mechanics, 301-346, Annual Review of Fluid Mechanics, 31. Annual Reviews, Palo Alto, Calif., 1999.
16. Dubreil-Jacotin, M.-L. Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie. J. Math. Pures Appl. 13 (1934), 217-291.
17. Gerstner, F. Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2 (1809), 412-445.
18. Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, 2001.
19. Gonzales, F.; Cokelet, E.; Gower, J.; Mulhern, M. SLAR and in-situ observations of wave-current interaction on the Columbia river bar. The ocean surface, 303-310. Reidel, New York, 1985.
20. Goyon, R. Contribution à la théorie des houles. Ann. Fac. Sci. Univ. Toulouse 22 (1958), 1-55.
21. Healey, T.; Kielhöfer, H. Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations. Arch. Rat. Mech. Anal. 113 (1991), 299-311.
22. Healey, T.; Simpson, H. Global continuation in nonlinear elasticity. Arch. Rat. Mech. Anal. 143 (1998), 1-28.
23. Johnson, R. S. A modern introduction to the mathematical theory of water waves. Cambridge Texts in Applied Mathematics. Cambridge University, Cambridge, 1997.
24. Jonsson, I. G. Wave-current interactions. The sea, 65-120. Ocean Engineering Science, 9. Wiley, New York, 1990.
25. Jonsson, I. G. Wave-current interactions. The sea, 65-120. Ocean Engineering Science, 9. Wiley, New York, 1990.
26. Kang, Y.; Vanden-Broeck, J.-M. Gravity-capillary waves in the presence of constant vorticity. Eur. J. Mech. B Fluids 19 (2000), no. 2, 253-268.
27. Keady, G.; Norbury, J. On the existence theory for irrotational water waves. Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 137-157.
28. Kemp, P. H.; Simons, R. R. The interaction between waves and a turbulent current: waves propagating with the current. J. Fluid Mech. 116 (1982), 227-250.
29. Kielhöfer, H. Multiple eigenvalue bifurcation for Fredholm operators. J. Reine Angew. Math. 358 (1985), 104-124.
30. Kishida, N.; Sobey, R. Stokes theory for waves on linear shear current. J. Engng. Mechanics 114 (1988), 1317-1334.
31. Krasovskiǐ, J. P. On the theory of steady-state waves of finite amplitude. Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961) 836-855. Translation in U.S.S.R. Comput. Math. and Math. Phys. 1 (1961), 996-1018.
32. Krasovskiǐ, J. P. On the theory of steady-state waves of finite amplitude. Ž. Vyčisl. Mat. i Mat. Fiz. 1 (1961) 836-855. Translation in U.S.S.R. Comput. Math. and Math. Phys. 1 (1961), 996-1018.
33. Levi-Civita, T. Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda. Rend. Accad. Lincei 33 (1924), 141-150.
34. Lieberman, G.; Trudinger, N. Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Amer. Math. Soc. 295 (1986), 509-546.
35. Lighthill, J. Waves in fluids. Cambridge University, Cambridge, 1978.
36. McLeod, J. B. The Stokes and Krasovskii conjectures for the wave of greatest height. University of Wisconsin Mathematics Research Center Report 2041, 1979.
37. McLeod, J. B. The Stokes and Krasovskii conjectures for the wave of greatest height. Stud. Appl. Math. 98 (1997), no. 4, 311-333.
38. Nekrasov, A. I. On steady waves. Izv. Ivanovo-Voznesenk. Politekhn. 3 (1921).
39. Okamoto, H.; Shōji, M. The mathematical theory of permanent progressive water-waves. Advanced Series in Nonlinear Dynamics, 20. World Scientific, River Edge, N.J., 2001.
40. Rabinowitz, P. Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7 (1971), 487-513.
41. Teles da Silva, A. F.; Peregrine, D. H. Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195 (1988), 281-302.
42. Serrin, J. A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), 304-318.
43. Stokes, G. On the theory of oscillatory waves. Trans. Cambridge Phil. Soc. 8 (1847), 441-455.
44. Struik, D. Détermination rigoureuse des ondes irrotationelles périodiques dans un canal à profondeur finie. Math. Ann. 95 (1926), 595-634.
45. Swan, C., Cummins, I.; James, R. An experimental study of two-dimensional surface water waves propagating on depth-varying currents. J. Fluid Mech. 428 (2001), 273-304.
46. Thomas, G. Wave-current interactions: an experimental and numerical study. Part 2. Nonlinear waves. J. Fluid Mech. 216 (1990), 505-536.
47. Thomas, G.; Klopman, G. Wave-current interactions in the nearshore region. Gravity waves in water of finite depth, 215-319. Advances in Fluid Mechanics, 10. WIT (Wessex Institute of Technology), Southhampton, United Kingdom, 1997.
48. Toland, J. F. On the existence of a wave of greatest height and Stokes's conjecture. Proc. Roy. Soc. London Ser. A 363 (1978), no. 1715, 469-485.
49. Toland, J. F. Stokes waves. Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1-48.
50. Vanden-Broeck, J.-M. Periodic waves with constant vorticity in water of infinite depth. IMA J. Appl. Math. 56 (1996), no. 3, 207-217.
51. Zeidler, E. Existenzbeweis für permanente Kapillar-Schwerewellen mit allgemeinen Wirbelverteilungen. Arch. Rational Mech. Anal. 50 (1973), 34-72.