Duke Mathematical Journal

Symmetry of steady periodic gravity water waves with vorticity

Adrian Constantin, Mats Ehrnström, and Erik Wahlén

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Abstract

We prove that steady periodic two-dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest, irrespective of the vorticity distribution within the fluid

Article information

Source
Duke Math. J., Volume 140, Number 3 (2007), 591-603.

Dates
First available in Project Euclid: 8 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1194547698

Digital Object Identifier
doi:10.1215/S0012-7094-07-14034-1

Mathematical Reviews number (MathSciNet)
MR2362244

Zentralblatt MATH identifier
1151.35076

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J60: Nonlinear elliptic equations 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

Citation

Constantin, Adrian; Ehrnström, Mats; Wahlén, Erik. Symmetry of steady periodic gravity water waves with vorticity. Duke Math. J. 140 (2007), no. 3, 591--603. doi:10.1215/S0012-7094-07-14034-1. https://projecteuclid.org/euclid.dmj/1194547698


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References

  • H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys. 5 (1988), 237--275.
  • A. Constantin, On the deep water wave motion, J. Phys. A 34 (2001), 1405--1417.
  • —, The trajectories of particles in Stokes waves, Invent. Math. 166 (2006), 523--535.
  • A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity, European J. Appl. Math. 15 (2004), 755--768.
  • —, Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech. 498 (2004), 171--181.
  • A. Constantin, D. Sattinger, and W. Strauss, Variational formulations for steady water waves with vorticity, J. Fluid. Mech. 548 (2006), 151--163.
  • A. Constantin and W. Strauss, Exact periodic traveling water waves with vorticity, C. R. Math. Acad. Sci. Paris 335 (2002), 797--800.
  • —, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math. 57 (2004), 481--527.
  • W. Craig and P. Sternberg, Symmetry of solitary waves, Comm. Partial Differential Equations 13 (1988), 603--633.
  • —, Symmetry of free-surface flows, Arch. Rational Mech. Anal. 118 (1992), 1--36.
  • M. EhrnströM, Uniqueness for steady periodic water waves with vorticity, Int. Math. Res. Not. 2005, no. 60, 3721--3726.
  • —, A note on surface profiles for symmetric gravity waves with vorticity, J. Nonlinear Math. Phys. 13 (2006), 1--8.
  • L. E. Fraenkel, Introduction to Maximum Principles and Symmetry in Elliptic Problems, Cambridge Tracts in Math. 128, Cambridge Univ. Press, Cambridge, 2000.
  • P. R. Garabedian, Surface waves of finite depth, J. Analyse Math. 14 (1965), 161--169.
  • F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile, Ann. Physik 32 (1809), 412--445.
  • B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209--243.
  • V. M. Hur, Symmetry of steady periodic water waves with vorticity, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 365 (2007), 2203--2214.
  • R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts Appl. Math., Cambridge Univ. Press, Cambridge, 1997.
  • J. Lighthill, Waves in Fluids, Cambridge Univ. Press, Cambridge, 1978.
  • H. Okamoto and M. ShōJi, The Mathematical Theory of Permanent Progressive Water-Waves, Adv. Ser. Nonlinear Dynam. 20, World Sci., River Edge, N.J., 2001.
  • W. J. M. Rankine, On the exact form of waves near the surface of deep water, Philos. Trans. Roy. Soc. London 153 (1863), 127--138.
  • C. Swan, I. P. Cummins, and R. L. James, An experimental study of two-dimensional surface water waves propagating on depth-varying currents, I: Regular waves, J. Fluid Mech. 428 (2001), 273--304.
  • J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal. 7 (1996), 1--48.
  • —, ``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 Monogr. Surv. Pure Appl. Math. 106, Chapman and Hall/CRC, Boca Raton, Fla., 2000, 207--217.
  • E. Varvaruca, Singularities of Bernoulli free boundaries, Comm. Partial Differential Equations 31 (2006), 1451--1477.
  • E. WahléN, A note on steady gravity waves with vorticity, Int. Math. Res. Not. 2005, no. 7, 389--396.
  • —, On rotational water waves with surface tension, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 365 (2007), 2215--2225.