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August  2014, 34(8): 3125-3133. doi: 10.3934/dcds.2014.34.3125

A characterization of the symmetric steady water waves in terms of the underlying flow

Bogdan-Vasile Matioc 1,

1. 

University of Vienna, Nordbergstraße 15, 1090, Vienna, Austria

Received  April 2013 Revised  September 2013 Published  January 2014

In this paper we present a characterization of the symmetric rotational periodic gravity water waves of finite depth and without stagnation points in terms of the underlying flow. Namely, we show that such a wave is symmetric and has a single crest and trough per period if and only if there exists a vertical line within the fluid domain such that all the fluid particles located on that line minimize there simultaneously their distance to the fluid bed as they move about. Our analysis uses the moving plane method, sharp elliptic maximum principles, and the principle of analytic continuation.
Keywords: Symmetry, underlying flow., gravity water waves, maximum principles.
Mathematics Subject Classification: Primary: 76B15; Secondary: 35Q31, 35B50, 26E0.
Citation: Bogdan-Vasile Matioc. A characterization of the symmetric steady water waves in terms of the underlying flow. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3125-3133. doi: 10.3934/dcds.2014.34.3125
References:
[1]

H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations,, J. Geom. Phys., 5 (1988), 237.  doi: 10.1016/0393-0440(88)90006-X.  Google Scholar

[2]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[3]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[4]

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

[5]

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

[6]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity,, European J. Appl. Math., 15 (2004), 755.  doi: 10.1017/S0956792504005777.  Google Scholar

[7]

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

[8]

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

[9]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133.  doi: 10.1007/s00205-011-0412-4.  Google Scholar

[10]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Rational Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[11]

W. Craig and P. Sternberg, Symmetry of solitary waves,, Comm. Partial Differential Equations, 13 (1988), 603.  doi: 10.1080/03605308808820554.  Google Scholar

[12]

J. Escher and B.-V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function,, Differential Integral Equations, ().   Google Scholar

[13]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer Verlag, (2001).   Google Scholar

[15]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[16]

V. M. Hur, Symmetry of solitary water waves with vorticity,, Math. Res. Lett., 15 (2008), 491.  doi: 10.4310/MRL.2008.v15.n3.a9.  Google Scholar

[17]

V. M. Hur, Symmetry of steady periodic water waves with vorticity,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2203.  doi: 10.1098/rsta.2007.2002.  Google Scholar

[18]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Univ. Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[19]

B. Kinsman, Wind Waves,, Prentice Hall, (1965).  doi: 10.1029/JZ066i008p02411.  Google Scholar

[20]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity,, Differential Integral Equations, 26 (2013), 129.   Google Scholar

[21]

A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves,, J. Evol. Equ., 12 (2012), 481.  doi: 10.1007/s00028-012-0141-7.  Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304.   Google Scholar

[23]

P. R. Garabedian, Surface waves of finite depth,, J. Anal. Math., 14 (1965), 161.  doi: 10.1007/BF02806385.  Google Scholar

[24]

J. F. Toland, On the symmetry theory for Stokes waves of finite and infinite depth,, in Trends in applications of mathematics to mechanics (Nice, (1998), 207.   Google Scholar

[25]

G. Tulzer, On the symmetry of steady periodic water waves with stagnation points,, Commun. Pure Appl. Anal., 11 (2012), 1577.  doi: 10.3934/cpaa.2012.11.1577.  Google Scholar

[26]

H. Okamoto and M. Shōji, The Mathematical Theory of Permanent Progressive Water-Waves,, Adv. Ser. Nonlinear Dynam., (2001).   Google Scholar

[27]

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

[28]

S. Walsh, Some criteria for the symmetry of stratified water waves,, Wave Motion, 46 (2009), 350.  doi: 10.1016/j.wavemoti.2009.06.008.  Google Scholar

show all references

References:
[1]

H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations,, J. Geom. Phys., 5 (1988), 237.  doi: 10.1016/0393-0440(88)90006-X.  Google Scholar

[2]

A. Constantin, On the deep water wave motion,, J. Phys. A, 34 (2001), 1405.  doi: 10.1088/0305-4470/34/7/313.  Google Scholar

[3]

A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis,, CBMS-NSF Conference Series in Applied Mathematics, (2011).  doi: 10.1137/1.9781611971873.  Google Scholar

[4]

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

[5]

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

[6]

A. Constantin and J. Escher, Symmetry of steady deep-water waves with vorticity,, European J. Appl. Math., 15 (2004), 755.  doi: 10.1017/S0956792504005777.  Google Scholar

[7]

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

[8]

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

[9]

A. Constantin and W. Strauss, Periodic traveling gravity water waves with discontinuous vorticity,, Arch. Ration. Mech. Anal., 202 (2011), 133.  doi: 10.1007/s00205-011-0412-4.  Google Scholar

[10]

A. Constantin and E. Varvaruca, Steady periodic water waves with constant vorticity: Regularity and local bifurcation,, Arch. Rational Mech. Anal., 199 (2011), 33.  doi: 10.1007/s00205-010-0314-x.  Google Scholar

[11]

W. Craig and P. Sternberg, Symmetry of solitary waves,, Comm. Partial Differential Equations, 13 (1988), 603.  doi: 10.1080/03605308808820554.  Google Scholar

[12]

J. Escher and B.-V. Matioc, On the analyticity of periodic gravity water waves with integrable vorticity function,, Differential Integral Equations, ().   Google Scholar

[13]

F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile,, Ann. Phys., 2 (1809), 412.   Google Scholar

[14]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer Verlag, (2001).   Google Scholar

[15]

D. Henry, On Gerstner's water wave,, J. Nonlinear Math. Phys., 15 (2008), 87.  doi: 10.2991/jnmp.2008.15.S2.7.  Google Scholar

[16]

V. M. Hur, Symmetry of solitary water waves with vorticity,, Math. Res. Lett., 15 (2008), 491.  doi: 10.4310/MRL.2008.v15.n3.a9.  Google Scholar

[17]

V. M. Hur, Symmetry of steady periodic water waves with vorticity,, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), 2203.  doi: 10.1098/rsta.2007.2002.  Google Scholar

[18]

R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge Univ. Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar

[19]

B. Kinsman, Wind Waves,, Prentice Hall, (1965).  doi: 10.1029/JZ066i008p02411.  Google Scholar

[20]

A.-V. Matioc and B.-V. Matioc, On the symmetry of periodic gravity water waves with vorticity,, Differential Integral Equations, 26 (2013), 129.   Google Scholar

[21]

A.-V. Matioc and B.-V. Matioc, Regularity and symmetry properties of rotational solitary water waves,, J. Evol. Equ., 12 (2012), 481.  doi: 10.1007/s00028-012-0141-7.  Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304.   Google Scholar

[23]

P. R. Garabedian, Surface waves of finite depth,, J. Anal. Math., 14 (1965), 161.  doi: 10.1007/BF02806385.  Google Scholar

[24]

J. F. Toland, On the symmetry theory for Stokes waves of finite and infinite depth,, in Trends in applications of mathematics to mechanics (Nice, (1998), 207.   Google Scholar

[25]

G. Tulzer, On the symmetry of steady periodic water waves with stagnation points,, Commun. Pure Appl. Anal., 11 (2012), 1577.  doi: 10.3934/cpaa.2012.11.1577.  Google Scholar

[26]

H. Okamoto and M. Shōji, The Mathematical Theory of Permanent Progressive Water-Waves,, Adv. Ser. Nonlinear Dynam., (2001).   Google Scholar

[27]

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

[28]

S. Walsh, Some criteria for the symmetry of stratified water waves,, Wave Motion, 46 (2009), 350.  doi: 10.1016/j.wavemoti.2009.06.008.  Google Scholar

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