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 & 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.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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.  doi: 10.1016/j.euromechflu.2010.09.008.  Google Scholar

[3]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, Journal of Fluid Mechanics, 498 (2004), 171.  doi: 10.1017/S0022112003006773.  Google Scholar

[4]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[5]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591.  doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

[6]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Communications on Pure and Applied Mathematics, 57 (2004), 481.  doi: 10.1002/cpa.3046.  Google Scholar

[7]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Philos. Trans. Roy. Soc. London A, 365 (2004), 2227.  doi: 10.1098/rsta.2007.2004.  Google Scholar

[8]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Communications on Pure and Applied Mathematics, 63 (2010), 533.  doi: 10.1002/cpa.20299.  Google Scholar

[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.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[10]

L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,", Cambridge University Press, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar

[11]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Communications in Mathematical Physics, 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[12]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,", Cambridge University Press, (1997).  doi: 10.2277/052159832X.  Google Scholar

[13]

J. Lighthill, "Waves in Fluids,", 2$nd$ edition, (2001).  doi: 10.2277/0521010454.  Google Scholar

[14]

H. Okamoto and M. Shoji, "The Mathematical Theory of Permanent Progressive Water-waves,", World Scientific, (2001).  doi: 10.1017/S0022112002002951.  Google Scholar

[15]

J. Serrin, A symmetry problem in potential theory,, Archive for Rational Mechanics and Analysis, 43 (1971), 304.  doi: 10.1007/BF00250468.  Google Scholar

[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.  doi: 10.1017/S0022112088002423.  Google Scholar

[17]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[18]

E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems,, Interfaces Free Bound., 9 (2007), 367.  doi: 10.4171/IFB/169.  Google Scholar

[19]

E. Varvaruca, On some properties of traveling water waves with vorticity,, SIAM J. Math. Anal., 39 (2008), 1686.  doi: 10.1137/070697513.  Google Scholar

[20]

E. Wahlén, Steady water waves with a critical layer,, Journal of Differential Equations, 246 (2009), 2468.  doi: 10.1016/j.jde.2008.10.005.  Google Scholar

show all references

References:
[1]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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.  doi: 10.1016/j.euromechflu.2010.09.008.  Google Scholar

[3]

A. Constantin and J. Escher, Symmetry of steady periodic surface water waves with vorticity,, Journal of Fluid Mechanics, 498 (2004), 171.  doi: 10.1017/S0022112003006773.  Google Scholar

[4]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559.  doi: 10.4007/annals.2011.173.1.12.  Google Scholar

[5]

A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity,, Duke Math. J., 140 (2007), 591.  doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar

[6]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity,, Communications on Pure and Applied Mathematics, 57 (2004), 481.  doi: 10.1002/cpa.3046.  Google Scholar

[7]

A. Constantin and W. Strauss, Rotational steady water waves near stagnation,, Philos. Trans. Roy. Soc. London A, 365 (2004), 2227.  doi: 10.1098/rsta.2007.2004.  Google Scholar

[8]

A. Constantin and W. Strauss, Pressure beneath a Stokes wave,, Communications on Pure and Applied Mathematics, 63 (2010), 533.  doi: 10.1002/cpa.20299.  Google Scholar

[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.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[10]

L. E. Fraenkel, "An Introduction to Maximum Principles and Symmetry in Elliptic Problems,", Cambridge University Press, (2000).  doi: 10.1017/CBO9780511569203.  Google Scholar

[11]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Communications in Mathematical Physics, 68 (1979), 209.  doi: 10.1007/BF01221125.  Google Scholar

[12]

R. S. Johnson, "A Modern Introduction to the Mathematical Theory of Water Waves,", Cambridge University Press, (1997).  doi: 10.2277/052159832X.  Google Scholar

[13]

J. Lighthill, "Waves in Fluids,", 2$nd$ edition, (2001).  doi: 10.2277/0521010454.  Google Scholar

[14]

H. Okamoto and M. Shoji, "The Mathematical Theory of Permanent Progressive Water-waves,", World Scientific, (2001).  doi: 10.1017/S0022112002002951.  Google Scholar

[15]

J. Serrin, A symmetry problem in potential theory,, Archive for Rational Mechanics and Analysis, 43 (1971), 304.  doi: 10.1007/BF00250468.  Google Scholar

[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.  doi: 10.1017/S0022112088002423.  Google Scholar

[17]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.   Google Scholar

[18]

E. Varvaruca, Some geometric and analytic properties of solutions of Bernoulli free-boundary problems,, Interfaces Free Bound., 9 (2007), 367.  doi: 10.4171/IFB/169.  Google Scholar

[19]

E. Varvaruca, On some properties of traveling water waves with vorticity,, SIAM J. Math. Anal., 39 (2008), 1686.  doi: 10.1137/070697513.  Google Scholar

[20]

E. Wahlén, Steady water waves with a critical layer,, Journal of Differential Equations, 246 (2009), 2468.  doi: 10.1016/j.jde.2008.10.005.  Google Scholar

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