Steady periodic waves bifurcating for fixed-depth rotational flows

Henry, David

doi:10.1090/S0033-569X-2013-01293-8

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
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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

T. Brooke Benjamin, The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech. 12 (1962), 97–116. MR 138286, DOI https://doi.org/10.1017/S0022112062000063
Boris Buffoni and John Toland, Analytic theory of global bifurcation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003. An introduction. MR 1956130
J. C. Burns, Long waves in running water, Proc. Cambridge Philos. Soc. 49 (1953), 695–706. With an appendix by M. J. Lighthill. MR 57668
G. R. Burton and J. F. Toland, Surface waves on steady perfect-fluid flows with vorticity, Comm. Pure Appl. Math. 64 (2011), no. 7, 975–1007. MR 2828587, DOI https://doi.org/10.1002/cpa.20365
Adrian Constantin, On the deep water wave motion, J. Phys. A 34 (2001), no. 7, 1405–1417. MR 1819940, DOI https://doi.org/10.1088/0305-4470/34/7/313
Adrian Constantin, Edge waves along a sloping beach, J. Phys. A 34 (2001), no. 45, 9723–9731. MR 1876166, DOI https://doi.org/10.1088/0305-4470/34/45/311
Adrian Constantin, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), no. 3, 523–535. MR 2257390, DOI https://doi.org/10.1007/s00222-006-0002-5
A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids 30 (2011), no. 1, 12–16. MR 2768364, DOI https://doi.org/10.1016/j.euromechflu.2010.09.008
Adrian Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. MR 2867413
Adrian Constantin, Mats Ehrnström, and Erik Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J. 140 (2007), no. 3, 591–603. MR 2362244, DOI https://doi.org/10.1215/S0012-7094-07-14034-1
Adrian Constantin and Joachim Escher, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech. 498 (2004), 171–181. MR 2256915, DOI https://doi.org/10.1017/S0022112003006773
Adrian Constantin and Joachim Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math. 15 (2004), no. 6, 755–768. MR 2144685, DOI https://doi.org/10.1017/S0956792504005777
Adrian Constantin and Joachim Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math. (2) 173 (2011), no. 1, 559–568. MR 2753609, DOI https://doi.org/10.4007/annals.2011.173.1.12
Adrian Constantin and Walter Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), no. 4, 481–527. MR 2027299, DOI https://doi.org/10.1002/cpa.3046
Adrian Constantin and Walter Strauss, Rotational steady water waves near stagnation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), no. 1858, 2227–2239. MR 2329144, DOI https://doi.org/10.1098/rsta.2007.2004
Adrian Constantin and Walter A. Strauss, Stability properties of steady water waves with vorticity, Comm. Pure Appl. Math. 60 (2007), no. 6, 911–950. MR 2306225, DOI https://doi.org/10.1002/cpa.20165
Adrian Constantin and Walter Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math. 63 (2010), no. 4, 533–557. MR 2604871, DOI https://doi.org/10.1002/cpa.20299
Adrian Constantin and Walter Strauss, Periodic traveling gravity water waves with discontinuous vorticity, Arch. Ration. Mech. Anal. 202 (2011), no. 1, 133–175. MR 2835865, DOI https://doi.org/10.1007/s00205-011-0412-4
Adrian Constantin and Eugen Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 33–67. MR 2754336, DOI https://doi.org/10.1007/s00205-010-0314-x
Adrian Constantin and Gabriele Villari, Particle trajectories in linear water waves, J. Math. Fluid Mech. 10 (2008), no. 1, 1–18. MR 2383410, DOI https://doi.org/10.1007/s00021-005-0214-2
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI https://doi.org/10.1016/0022-1236%2871%2990015-2
A. F. Teles 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, DOI https://doi.org/10.1017/S0022112088002423

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-London, 1972. Probability and Mathematical Statistics, No. 14. MR 0442564

Mats Ehrnström and Gabriele Villari, Linear water waves with vorticity: rotational features and particle paths, J. Differential Equations 244 (2008), no. 8, 1888–1909. MR 2409513, DOI https://doi.org/10.1016/j.jde.2008.01.012

Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845

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.

Mariano Giaquinta and Stefan Hildebrandt, Calculus of variations. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 310, Springer-Verlag, Berlin, 1996. The Lagrangian formalism. MR 1368401

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364

David Henry, The trajectories of particles in deep-water Stokes waves, Int. Math. Res. Not. , posted on (2006), Art. ID 23405, 13. MR 2272104, DOI https://doi.org/10.1155/IMRN/2006/23405

David Henry, Particle trajectories in linear periodic capillary and capillary-gravity water waves, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), no. 1858, 2241–2251. MR 2329145, DOI https://doi.org/10.1098/rsta.2007.2005

David Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys. 15 (2008), no. suppl. 2, 87–95. MR 2434727, DOI https://doi.org/10.2991/jnmp.2008.15.s2.7

David Henry, Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity, SIAM J. Math. Anal. 42 (2010), no. 6, 3103–3111. MR 2763714, DOI https://doi.org/10.1137/100801408

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, Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996. With applications to Schrödinger operators. MR 1361167

R. S. Johnson, A modern introduction to the mathematical theory of water waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1997. MR 1629555

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.

Joy Ko and Walter Strauss, Large-amplitude steady rotational water waves, Eur. J. Mech. B Fluids 27 (2008), no. 2, 96–109. MR 2389493, DOI https://doi.org/10.1016/j.euromechflu.2007.04.004

Joy Ko and Walter Strauss, Effect of vorticity on steady water waves, J. Fluid Mech. 608 (2008), 197–215. MR 2439751, DOI https://doi.org/10.1017/S0022112008002371

James Lighthill, Waves in fluids, Cambridge University Press, Cambridge-New York, 1978. MR 642980

M. Lilli and J. F. Toland, Waves on a steady stream with vorticity, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 267–277. MR 2500496, DOI https://doi.org/10.1090/pspum/079/2500496

Bogdan-Vasile Matioc, Analyticity of the streamlines for periodic traveling water waves with bounded vorticity, Int. Math. Res. Not. IMRN 17 (2011), 3858–3871. MR 2836396, DOI https://doi.org/10.1093/imrn/rnq235

B. V. Matioc, On the regularity of deep-water waves with general vorticity distributions, Quart. Appl. Math. 70 (2012), 393–405.

Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959

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.

Erik Wahlén, Steady water waves with a critical layer, J. Differential Equations 246 (2009), no. 6, 2468–2483. MR 2498849, DOI https://doi.org/10.1016/j.jde.2008.10.005

Erik Wahlén, Steady periodic capillary-gravity waves with vorticity, SIAM J. Math. Anal. 38 (2006), no. 3, 921–943. MR 2262949, DOI https://doi.org/10.1137/050630465

S. Walsh, Steady periodic gravity waves with surface tension, preprint.

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1–48. MR 1422004, DOI https://doi.org/10.12775/TMNA.1996.001