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

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.
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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[22]

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

[23]

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

[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.

[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.

[26]

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

[27]

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

[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.

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