# American Institute of Mathematical Sciences

July  2012, 11(4): 1577-1586. doi: 10.3934/cpaa.2012.11.1577

## On the symmetry of steady periodic water waves with stagnation points

 1 Faculty of Mathematics, University of Vienna, Nordbergstraße 15, 1090 Vienna, Austria

Received  June 2011 Revised  September 2011 Published  January 2012

The aim of this paper is to prove the symmetry of small-amplitude steady periodic water waves with monotonic wave profile even if stagnation points occur in the flow beneath the wave.
Citation: Gerhard Tulzer. On the symmetry of steady periodic water waves with stagnation points. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1577-1586. doi: 10.3934/cpaa.2012.11.1577
##### References:
 [1] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [2] A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008. [3] A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, Journal of Fluid Mechanics, 498 (2004), 171-181. doi: 10.1017/S0022112003006773. [4] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [5] A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1. [6] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Communications on Pure and Applied Mathematics, 57 (2004), 481-507. doi: 10.1002/cpa.3046. [7] A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London A, 365 (2004), 2227-2239. doi: 10.1098/rsta.2007.2004. [8] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Communications on Pure and Applied Mathematics, 63 (2010), 533-557. doi: 10.1002/cpa.20299. [9] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Archive for Rational Mechanics and Analysis, 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x. [10] L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge University Press, 2000. doi: 10.1017/CBO9780511569203. [11] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Communications in Mathematical Physics, 68 (1979), 209-283. doi: 10.1007/BF01221125. [12] R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves," Cambridge University Press, Cambridge, 1997. doi: 10.2277/052159832X. [13] J. Lighthill, "Waves in Fluids," 2$nd$ edition, Cambridge University Press, 2001. doi: 10.2277/0521010454. [14] H. Okamoto and M. Shoji, "The Mathematical Theory of Permanent Progressive Water-waves," World Scientific, River Edge, N.J., 2001. doi: 10.1017/S0022112002002951. [15] J. Serrin, A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis, 43 (1971), 304-318. doi: 10.1007/BF00250468. [16] 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. doi: 10.1017/S0022112088002423. [17] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. [18] E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems, Interfaces Free Bound., 9 (2007), 367-381. doi: 10.4171/IFB/169. [19] E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal., 39 (2008), 1686-1692. doi: 10.1137/070697513. [20] E. Wahlén, Steady water waves with a critical layer, Journal of Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.

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##### References:
 [1] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5. [2] A. Constantin, Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train, Eur. J. Mech. B Fluids, 30 (2011), 12-16. doi: 10.1016/j.euromechflu.2010.09.008. [3] A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity, Journal of Fluid Mechanics, 498 (2004), 171-181. doi: 10.1017/S0022112003006773. [4] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. of Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [5] A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1. [6] A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Communications on Pure and Applied Mathematics, 57 (2004), 481-507. doi: 10.1002/cpa.3046. [7] A. Constantin and W. Strauss, Rotational steady water waves near stagnation, Philos. Trans. Roy. Soc. London A, 365 (2004), 2227-2239. doi: 10.1098/rsta.2007.2004. [8] A. Constantin and W. Strauss, Pressure beneath a Stokes wave, Communications on Pure and Applied Mathematics, 63 (2010), 533-557. doi: 10.1002/cpa.20299. [9] A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: regularity and local bifurcation, Archive for Rational Mechanics and Analysis, 199 (2011), 33-67. doi: 10.1007/s00205-010-0314-x. [10] L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems," Cambridge University Press, 2000. doi: 10.1017/CBO9780511569203. [11] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Communications in Mathematical Physics, 68 (1979), 209-283. doi: 10.1007/BF01221125. [12] R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves," Cambridge University Press, Cambridge, 1997. doi: 10.2277/052159832X. [13] J. Lighthill, "Waves in Fluids," 2$nd$ edition, Cambridge University Press, 2001. doi: 10.2277/0521010454. [14] H. Okamoto and M. Shoji, "The Mathematical Theory of Permanent Progressive Water-waves," World Scientific, River Edge, N.J., 2001. doi: 10.1017/S0022112002002951. [15] J. Serrin, A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis, 43 (1971), 304-318. doi: 10.1007/BF00250468. [16] 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. doi: 10.1017/S0022112088002423. [17] J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. [18] E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems, Interfaces Free Bound., 9 (2007), 367-381. doi: 10.4171/IFB/169. [19] E. Varvaruca, On some properties of traveling water waves with vorticity, SIAM J. Math. Anal., 39 (2008), 1686-1692. doi: 10.1137/070697513. [20] E. Wahlén, Steady water waves with a critical layer, Journal of Differential Equations, 246 (2009), 2468-2483. doi: 10.1016/j.jde.2008.10.005.
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