Deep-water waves with vorticity: Symmetry and rotational behaviour

Discrete and Continuous Dynamical Systems

Volumn 19, Issue 3, 2007, Pages 483-491

  Ehrnström, Mats ^a  

^a LUND UNIVERSITY   (Sweden)

Author keywords

Nonlinear partial differential equation; Symmetry; Vorticity; Water waves

Indexed keywords

EID: 37349095016     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2007.19.483     Document Type: Conference Paper

References (28)

1. J. Albert and F. Linares, Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves, J. Math. Pures Appl., 79 (2000), 195-226.
2. H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.
3. A. Constantin, On the deep water wave motion, J. Phys. A, 34 (2001), 1405-1417.
4. A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.
5. A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), 171-181.
6. A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), 755-768.
7. A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.
8. A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, To appear in Duke Math. J.
9. A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527.
10. W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations, 13 (1988), 603-633.
11. W. Craig and P. Sternberg, Symmetry of free-surface flows, Arch. Rational Mech. Anal., 118 (1992), 1-36.
12. M. L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), 217-291.
13. M. Ehrnström, Uniqueness for steady periodic water waves with vorticity, Int. Math. Res. Not., 60 (2005), 3721-3726.
14. M. Ehrnström, A Note on Surface Profiles for Symmetric Gravity Waves with Vorticity, J. Nonlinear Math. Phys., 13 (2006), 1-8.
15. L. E. Fraenkel, "Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge University Press, Cambridge, 2000.
16. P. R. Garabedian, Surface waves of finite depth, J. Analyse Math., 14 (1965), 161-169.
17. F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys., 2 (1809), 412-445.
18. B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.
19. E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vomelliptischen Typus, Sitzungsberichte Preussiche Akademie Wissenschaften, Berlin (1927) 147-152.
20. V. M. Hur, Symmetry of steady periodic water waves with vorticity, To appear in Philos. Trans. R. Soc. Lond. Ser. A.
21. V. M. Hur, Global bifurcation theory of deep-water waves with vorticity, SIAM J. Math. Anal., 37 (2006), 1482-1521.
22. L. Lighthill, "Waves in Fluids," Cambridge University Press, 1978.
23. L. A. Maia, Symmetry of internal waves, Nonlinear Anal., 28 (1997), 87-102.
24. H. Okamoto and M. Shōji, "The Mathematical Theory of Permanent Progressive Waterwaves," World Sci. Publishing, River Edge, NJ, 2001.
25. J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.
26. J. F. Toland, On the symmetry theory for Stokes waves of finite and infinite depth, in "Trends in Applications of Mathematics to Mechanics (Nice, 1998)," Chapman & Hall/CRC, Boca Raton, FL.
27. E. Varvaruca, Singularities of Bernoulli free boundaries, Comm. Partial Differential Equations, 31 (2006), 1451-1477.
28. E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal., 38 (2006), 921-943.