Steady periodic waves bifurcating for fixed-depth rotational flows
Author:
David Henry
Journal:
Quart. Appl. Math. 71 (2013), 455-487
MSC (2010):
Primary 35B32, 35Q31, 35J25
DOI:
https://doi.org/10.1090/S0033-569X-2013-01293-8
Published electronically:
May 16, 2013
MathSciNet review:
3112823
Full-text PDF Free Access
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Additional Information
Abstract: We consider steady periodic water waves for rotational flows with a specified fixed depth over a flat bed. We construct a modified height function, which explicitly introduces the mean depth into the rotational water wave problem, and use the Crandall-Rabinowitz local bifurcation theorem to establish the existence of solutions of the resulting problem.
References
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References
- T. B. Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech. 12 (1962), 97–116. MR 0138286 (25:1733)
- B. Buffoni and J. F. Toland, Analytic theory of global bifurcation, Princeton University Press, Princeton and Oxford, 2003. MR 1956130 (2004b:47117)
- J. C. Burns, Long waves on running water, Proc. Cambridge Phil. Soc. 49 (1953), 695–706. MR 0057668 (15:261a)
- G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math. 64 (2011), 975–1007. MR 2828587 (2012e:74063)
- A. Constantin, On the deep water wave motion, J. Phys. A 34 (2001), 1405–1417. MR 1819940 (2002b:76010)
- A. Constantin, Edge waves along a sloping beach, J. Phys. A 34 (2001), 9723–9731. MR 1876166 (2002j:76015)
- A. Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), 523–535. MR 2257390 (2007j:35240)
- A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Europ. J. Mech. B Fluids 30 (2011), 12–16. MR 2768364
- A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, CBMS-NSF Conference Series in Applied Mathematics, Vol. 81, SIAM, Philadelphia, PA, 2011. MR 2867413
- A. Constantin, M. Ehrnström, and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), 591–603. MR 2362244 (2009c:35359)
- A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech. 498 (2004), 171–181. MR 2256915 (2007f:76023)
- A. Constantin and J. Escher, Symmetry of deep-water waves with vorticity, European J. Appl. Math. 15 (2004), 755–768. MR 2144685 (2006b:76013)
- A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2) 173 (2011), 559–568. MR 2753609
- A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), 481–527. MR 2027299 (2004i:76018)
- A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. R. Soc. Lond. Ser. A 365 (2007), 2227–2239. MR 2329144 (2008j:76013)
- A. Constantin and W. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. 60 (2007), 911–950. MR 2306225 (2009b:35257)
- A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 53 (2010), 533–557. MR 2604871 (2011b:76017)
- A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Rational Mech. Anal. 202 (2011), 133–175. MR 2835865 (2012g:76023)
- A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Rational Mech. Anal. 199 (2011), 33–67. MR 2754336
- A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech. 10 (2008), 1–18. MR 2383410 (2009a:76017)
- M. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. MR 0288640 (44:5836)
- A. F. T. da Silva and D. H. Peregrine, Steep, steady surface waves on water of finite depth with constant vorticity, J. Fluid Mech. 195 (1988), 281–302. MR 985439 (90a:76061)
- M.-L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes periodiques d’ampleur finie, J. Math. Pures Appl. 13 (1934), 217–291.
- H. Dym and H. P. McKean, Fourier Series and Integrals, Academic Press, New York, 1972. MR 0442564 (56:945)
- M. Ehrnström and G. Villari, Linear water waves with vorticity: Rotational features and particle paths, J. Differential Equations 244 (2008), 1888–1909. MR 2409513 (2009i:76026)
- L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, Rhode Island, 1998. MR 1625845 (99e:35001)
- N.C. Freeman and R.S. Johnson, Shallow water waves on shear flows, J. Fluid Mech. 42 (1970), 401–409.
- F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Phys. 2 (1809), 412–445.
- M. Giaquinta and S. Hildebrandt, Calculus of variations. I. The Lagrangian formalism, Grundlehren der Mathematischen Wissenschaften, 310. Springer-Verlag, Berlin, 1996. MR 1368401 (98b:49002a)
- D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 2001. MR 1814364 (2001k:35004)
- D. Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. (2006), Art. ID 23405, 13 pp. MR 2272104 (2007k:76017)
- D. Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Phil. Trans. Roy. Soc. A, 365 (2007), 2241–2251. MR 2329145 (2008i:76023)
- D. Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), 87–95. MR 2434727 (2009k:76020)
- D. Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal. 42 (2010), 3103–3111. MR 2763714
- D. Henry, Analyticity of the free surface for periodic travelling capillary-gravity water waves with vorticity, J. Math. Fluid Mech. 14 (2012), 249–254.
- P. D. Hislop and I. M. Sigal, Introduction to spectral theory. With applications to Schrödinger operators, Applied Mathematical Sciences, 113. Springer-Verlag, New York, 1996. MR 1361167 (98h:47003)
- R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Univ. Press, Cambridge, 1997. MR 1629555 (99m:76017)
- I. G. Jonsson, Wave-current interactions, The sea, 65–120. Ocean Engineering Science, 9. Wiley, New York, 1990.
- B. Kinsman, Wind Waves, Prentice Hall, New Jersey, 1965.
- J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Europ. J. Mech. B Fluids 27 (2007), 96–109. MR 2389493 (2009b:76014)
- J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech. 608 (2008), 197–215. MR 2439751 (2009f:76023)
- J. Lighthill, Waves in fluids, Cambridge University Press, Cambridge, 1978. MR 642980 (84g:76001a)
- M. Lilli and J. F. Toland, Waves on a steady stream with vorticity, Perspectives in partial differential equations, harmonic analysis and applications, 267–277, Proc. Sympos. Pure Math., 79, Amer. Math. Soc., Providence, RI, 2008. MR 2500496 (2010i:35447)
- B. V. Matioc, Analyticity of the streamlines for periodic traveling water waves with bounded vorticity, Int. Math. Res. Not. 2011 (2011), 3858–3871. MR 2836396
- B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math. 70 (2012), 393–405.
- M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, New York, 1980. MR 751959 (85e:46002)
- G. Thomas and G. Klopman, Wave-current interactions in the nearshore region, Gravity waves in water of finite depth, 215–319, Advances in Fluid Mechanics, 10. WIT, Southhampton, United Kingdom, 1997.
- E. Wahlén, Steady water waves with a critical layer, J. Differential Equations 246 (2009), 2468–2483. MR 2498849 (2010i:76026)
- E. Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal. 38 (2006), 921–943. MR 2262949 (2007g:35196)
- S. Walsh, Steady periodic gravity waves with surface tension, preprint.
- J. F. Toland, Stokes waves, Topol. Meth. Nonl. Anal. 7 (1996), 1–48. MR 1422004 (97j:35130)
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Additional Information
David Henry
Affiliation:
School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
MR Author ID:
766545
Email:
david.henry@dcu.ie
Keywords:
Local-bifurcation,
steady periodic waves,
vorticity,
fixed-depth flows.
Received by editor(s):
June 26, 2011
Published electronically:
May 16, 2013
Article copyright:
© Copyright 2013
Brown University