Steady periodic capillary-gravity waves with vorticity

SIAM Journal on Mathematical Analysis

Volumn 38, Issue 3, 2006, Pages 921-943

  Wahlén, Erik ^a  

^a LUND UNIVERSITY   (Sweden)

Author keywords

Bifurcation theory; Capillarity; Vorticity; Water waves

Indexed keywords

BIFURCATION (MATHEMATICS); BOUNDARY CONDITIONS; MATHEMATICAL TRANSFORMATIONS; VORTICITY;

CAPILLARY-GRAVITY WAVES; SCHAUDER THEORY; VENTTSEL BOUNDARY CONDITIONS; VORTICITY DISTRIBUTION;

GRAVITY WAVES;

EID: 33947375479     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/050630465     Document Type: Article

References (41)

1. J. BOGNÁR, Indefinite Inner Product Spaces, Springer-Verlag, New York, Heidelberg, 1974.
2. A. CONSTANTIN, On the deep water wave motion, J. Phys. A, 34 (2001), pp. 1405-1417.
3. A. CONSTANTIN, Edge waves along a sloping beach, J. Phys. A, 34 (2001), pp. 9723-9731.
4. A. CONSTANTIN AND J. ESCHER, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech., 498 (2004), pp. 171-181.
5. A. CONSTANTIN AND J. ESCHER, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math., 15 (2004), pp. 755-768.
6. A. CONSTANTIN AND W. STRAUSS, Exact periodic traveling water waves with vorticity, C. R. Math. Acad. Sci. Paris, 335 (2002), pp. 797-800.
7. A. CONSTANTIN AND W. STRAUSS, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), pp. 481-527.
8. W. CRAIG AND D. P. NICHOLLS, Traveling two and three dimensional capillary gravity water waves, SIAM J. Math. Anal., 32 (2000), pp. 323-359.
9. M. CRANDALL AND P. RABINOWITZ, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), pp. 321-340.
10. G. D. CRAPPER, An exact solution for progressive capillary waves of arbitrary amplitude, J. Fluid Mech., 2 (1957), pp. 532-540.
11. G. D. CRAPPER, Introduction to Water Waves, Ellis Horwood, Chichester, Halsted Press, New York, 1984.
12. F. DIAS AND C. KHARIF, Nonlinear gravity and capillary-gravity waves, in Annual Review of Fluid Mechanics, Vol. 31, Annual Reviews, Palo Alto, CA, 1999, pp. 301-346.
13. M.-L. DUBREIL-JACOTIN, Sur la determination rigoureuse des ondes permanentes périodiques d'ampleur finie, J. Math. Pures Appl., 13 (1934), pp. 217-291.
14. M. EHRNSTRÖM, Uniqueness of steady symmetric deep-water waves with vorticity, J. Nonlinear Math. Phys., 12 (2005), pp. 27-30.
15. D. GILBARG AND N. S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
16. M. D. GROVES, Steady water waves, J. Nonlinear Math. Phys., 11 (2004), pp. 435-460.
17. L. HÖRMANDER, Fourier integral operators. I, Acta Math., 127 (1971), pp. 79-183.
18. I. S. IOHVIDOV, M. G. KREIN, AND H. LANGER, Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Akademie-Verlag, Berlin, 1982.
19. R. S. JOHNSON, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge University Press, Cambridge, UK, 1997.
20. M. JONES, Small amplitude capillary-gravity waves in a channel of finite depth, Glasgow Math. J., 31 (1989), pp. 141-160.
21. M. JONES AND J. TOLAND, The bifurcation and secondary bifurcation of capillary-gravity waves, Proc. Roy. Soc. London Ser. A, 399 (1985), pp. 391-417.
22. M. JONES AND J. TOLAND, Symmetry and the bifurcation of capillary-gravity waves, Arch. Rational Mech. Anal., 96 (1986), pp. 29-53.
23. H. KALISCH, Periodic traveling water waves with isobaric streamlines, J. Nonlinear Math. Phys., 11 (2004), pp. 461-471.
24. H. KALISCH, A uniqueness result for periodic traveling waves in water of finite depth, Nonlinear Anal., 58 (2004), pp. 779-785.
25. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1976.
26. W. KINNERSLEY, Exact large amplitude capillary waves on sheets of fluid, J. Fluid Mech., 77 (1976), pp. 229-241.
27. H. LAMB, Hydrodynamics, Cambridge University Press, London, 1924.
28. J. LIGHTHILL, Waves in Fluids, Cambridge University Press, Cambridge, UK, 1978.
29. Y. LUO AND N. TRUDINGER, Linear second order elliptic equations with Venttsel boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 118 (1991), pp. 193-207.
30. L. NIRENBERG, Topics in Nonlinear Functional Analysis, AMS, Providence, RI, 2001.
31. D. P. NICHOLLS AND F. REITICH, On analyticity of traveling water waves, Proc. Roy. Soc. London Ser. A, 461 (2005), pp. 1283-1309.
32. H. OKAMOTO AND M. SHOJI, The Mathematical Theory of Permanent Progressive Water-Waves, World Scientific, River Edge, NJ, 2001.
33. J. REEDER AND M. SHINBROT, On Wilton ripples. I. Formal derivation of the phenomenon, Wave Motion, 3 (1981), pp. 115-135.
34. J. REEDER AND M. SHINBROT, On Wilton ripples. II. Rigorous results, Arch. Rational Mech. Anal., 77 (1981), pp. 321-347.
35. T. DA SILVA AND D. H. PEREGRINE, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech., 195 (1988), pp. 281-302.
36. C. SWAN, I. CUMMINS, AND R. JAMES, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, J. Fluid Mech., 428 (2001), pp. 273-304.
37. G. THOMAS AND G. KLOPMAN, Wave-current interactions in the nearshore region, in Gravity Waves in Water of Finite Depth, Adv. Fluid Mech. 10, Computational Mechanics Publications, Southampton, UK, 1997, pp. 215-319.
38. J. F. TOLAND, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), pp. 1-48.
39. E. WAHLEN, Uniqueness for autonomous planar differential equations and the Lagrangian formulation of water flows with vorticity, J. Nonlinear Math. Phys., 11 (2004), pp. 549-555.
40. E. WAHLEN, A note on steady gravity waves with vorticity, Int. Math. Res. Not., 7 (2005), pp. 389-396.
41. E. WAHLEN, Steady periodic capillary waves with vorticity, Ark. Mat., to appear.