Steady water waves with a critical layer

Journal of Differential Equations

Volumn 246, Issue 6, 2009, Pages 2468-2483

  Wahlén, Erik ^a  

^a LUND UNIVERSITY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 59649099110     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2008.10.005     Document Type: Article

References (45)

1. Benney D., and Bergeron R. A new class of nonlinear waves in parallel flows. Stud. Appl. Math. 48 (1969) 181-204
2. Bridges T.J., and Donaldson N.M. Secondary criticality of water waves. I. Definition, bifurcation and solitary waves. J. Fluid Mech. 565 (2006) 381-417
3. Buffoni B., and Toland J. Analytic Theory of Global Bifurcation. Princeton Ser. Appl. Math. (2003), Princeton Univ. Press, Princeton, NJ an introduction
4. Constantin A. The trajectories of particles in Stokes waves. Invent. Math. 166 (2006) 523-535
5. Constantin A., Ehrnström M., and Villari G. Particle trajectories in linear deep-water waves. Nonlinear Anal. Real World Appl. 9 (2008) 1336-1344
6. Constantin A., Ehrnström M., and Wahlén E. Symmetry for steady gravity water waves with vorticity. Duke Math. J. 140 (2007) 591-603
7. Constantin A., and Escher J. Symmetry of steady deep-water waves with vorticity. European J. Appl. Math. 15 (2004) 755-768
8. Constantin A., and Escher J. Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech. 498 (2004) 171-181
9. Constantin A., and Escher J. Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. (N.S.) 44 (2007) 423-431
10. Constantin A., Sattinger D., and Strauss W. Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548 (2006) 151-163
11. Constantin A., and Strauss W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004) 481-527
12. Constantin A., and Strauss W. Stability properties of steady water waves with vorticity. Comm. Pure Appl. Math. 60 (2007) 911-950
13. Constantin A., and Villari G. Particle trajectories in linear water waves. J. Math. Fluid Mech. 10 (2008) 1-18
14. Crandall M.G., and Rabinowitz P.H. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971) 321-340
15. Davis R. On the high Reynolds number flow over a wavy boundary. J. Fluid Mech. 36 (1969) 337-346
16. Dubreil-Jacotin M.L. Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finite. J. Math. Pures Appl. 13 (1934) 217-291
17. Ehrnström M. Uniqueness for steady periodic water waves with vorticity. Int. Math. Res. Not. (2005) 3721-3726
18. Ehrnström M. Uniqueness of steady symmetric deep-water waves with vorticity. J. Nonlinear Math. Phys. 12 (2005) 27-30
19. Ehrnström M. Deep-water waves with vorticity: Symmetry and rotational behaviour. Discrete Contin. Dyn. Syst. 19 (2007) 483-491
20. Ehrnström M. A uniqueness result for steady symmetric water waves with affine vorticity. Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal. 14 (2007)
21. Ehrnström M. On the streamlines and particle paths of gravitational water waves. Nonlinearity 21 (2008) 1-14
22. Ehrnström M., and Villari G. Linear water waves with vorticity: Rotational features and particle paths. J. Differential Equations 244 (2008) 1888-1909
23. Gilbarg D., and Trudinger N.S. Elliptic Partial Differential Equations of Second Order. Classics Math. (2001), Springer-Verlag, Berlin reprint of the 1998 edition
24. Groves M.D. Steady water waves. J. Nonlinear Math. Phys. 11 (2004) 435-460
25. Groves M.D., and Mielke A. A spatial dynamics approach to three-dimensional gravity-capillary steady water waves. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 83-136
26. Groves M.D., and Wahlén E. Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity. SIAM J. Math. Anal. 39 (2007) 932-964 (electronic)
27. Groves M.D., and Wahlén E. Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity. Phys. D 237 (2008) 1530-1538
28. Henry D. The trajectories of particles in deep-water Stokes waves. Int. Math. Res. Not. 2006 (2006) 1-13 10.1155/IMRN/2006/23405
29. Henry D. Particle trajectories in linear periodic capillary and capillary-gravity deep-water waves. J. Nonlinear Math. Phys. 14 (2007) 1-7
30. Hur V.M. Global bifurcation theory of deep-water waves with vorticity. SIAM J. Math. Anal. 37 (2006) 1482-1521 (electronic)
31. Hur V.M. Symmetry of steady periodic water waves with vorticity. Philos. Trans. R. Soc. Lond. Ser. A 365 (2007) 2203-2214
32. Hur V.M. Exact solitary water waves with vorticity. Arch. Ration. Mech. Anal. 188 (2008) 213-244
33. Johnson R.S. On the nonlinear critical layer below a nonlinear unsteady surface wave. J. Fluid Mech. 167 (1986) 327-351
34. Johnson R.S. A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge Texts Appl. Math. (1997), Cambridge Univ. Press, Cambridge
35. Kalisch H. A uniqueness result for periodic traveling waves in water of finite depth. Nonlinear Anal. 58 (2004) 779-785
36. Ko J., and Strauss W. Large-amplitude steady rotational water waves. Eur. J. Mech. B Fluids 27 (2008) 96-109
37. Lighthill J. Waves in Fluids (1978), Cambridge Univ. Press, Cambridge
38. M. Lilli, J.F. Toland, Waves on a steady stream with vorticity, preprint
39. Moore D., and Saffman P. Finite-amplitude waves in inviscid shear flows. Proc. R. Soc. Lond. Ser. A 382 (1982) 389-410
40. Redekopp L.G. On the theory of solitary Rossby waves. J. Fluid Mech. 82 (1977) 725-745
41. Stokes G.G. On the theory of oscillatory waves. Trans. Cambridge Philos. Soc. 8 (1849) 441-455
42. Stuart J. On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29 (1967) 417-440
43. Toland J.F. Stokes waves. Topol. Methods Nonlinear Anal. 7 [8] (1996 [1997]) 1-48 [412-414]
44. Wahlén E. Hamiltonian long-wave approximations of water waves with constant vorticity. Phys. Lett. A 372 (2008) 2597-2602
45. Wahlén E. Steady periodic capillary-gravity waves with vorticity. SIAM J. Math. Anal. 38 (2006) 921-943 (electronic)