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June  2012, 17(4): 1101-1112. doi: 10.3934/dcdsb.2012.17.1101

Vorticity jumps in steady water waves

Walter A. Strauss 1,

1. 

Brown University, Department of Mathematics and Lefschetz Center for Dynamical Systems, Providence, RI 02912, United States

Received  December 2010 Revised  September 2011 Published  February 2012

There exists a large family of water waves with jump discontinuities in the vorticity. These waves travel at a constant speed. They are two-dimensional, periodic, symmetric, and subject to the influence of gravity. Some of them have large amplitudes. Their existence is proven using local and global bifurcation theory, together with elliptic theory of weak solutions with nonlinear boundary conditions.
Keywords: vorticity, nonlinear elliptic., Water waves.
Mathematics Subject Classification: Primary: 76B15, 35J66; Secondary: 35Q3.
Citation: Walter A. Strauss. Vorticity jumps in steady water waves. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1101-1112. doi: 10.3934/dcdsb.2012.17.1101
References:
[1]

J. T. Beale, The existence of solitary water waves, Comm. Pure Appl. Math., 30 (1977), 373-389. doi: 10.1002/cpa.3160300402.

[2]

B. Buffoni and J. F. Toland, "Analytic Theory of Global Bifurcation. An Introduction," Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003.

[3]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[4]

A. Constantin and W. Strauss, Periodic traveling gravity waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[5]

R. Finn and D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63.

[6]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[7]

T. Healey and H. Simpson, Global continuation in nonlinear elasticity, Arch. Ration. Mech. Anal., 143 (1998), 1-28. doi: 10.1007/s002050050098.

[8]

G. M. Lieberman, The nonlinear oblique derivative problem for quasilinear elliptic equations, Nonlinear Anal., 8 (1984), 49-65. doi: 10.1016/0362-546X(84)90027-0.

[9]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Europ. J. Mech. B Fluids, 27 (2007), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.

[10]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371.

[11]

O. M. Philllips and M. L. Banner, Wave breaking in presence of wind drift and swell, J. Fluid Mech., 66 (1974), 625-640. doi: 10.1017/S0022112074000413.

[12]

J. Serrin, A symmetry theorem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318. doi: 10.1007/BF00250468.

[13]

W. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694.

[14]

S. Walsh, Stratified and steady periodic gravity waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[15]

S. Walsh, Steady periodic gravity waves with surface tension, preprint, arXiv:0911.1375.

show all references

References:
[1]

J. T. Beale, The existence of solitary water waves, Comm. Pure Appl. Math., 30 (1977), 373-389. doi: 10.1002/cpa.3160300402.

[2]

B. Buffoni and J. F. Toland, "Analytic Theory of Global Bifurcation. An Introduction," Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003.

[3]

A. Constantin and W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 57 (2004), 481-527. doi: 10.1002/cpa.3046.

[4]

A. Constantin and W. Strauss, Periodic traveling gravity waves with discontinuous vorticity, Arch. Ration. Mech. Anal., 202 (2011), 133-175. doi: 10.1007/s00205-011-0412-4.

[5]

R. Finn and D. Gilbarg, Asymptotic behavior and uniqueness of plane subsonic flows, Comm. Pure Appl. Math., 10 (1957), 23-63.

[6]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[7]

T. Healey and H. Simpson, Global continuation in nonlinear elasticity, Arch. Ration. Mech. Anal., 143 (1998), 1-28. doi: 10.1007/s002050050098.

[8]

G. M. Lieberman, The nonlinear oblique derivative problem for quasilinear elliptic equations, Nonlinear Anal., 8 (1984), 49-65. doi: 10.1016/0362-546X(84)90027-0.

[9]

J. Ko and W. Strauss, Large-amplitude steady rotational water waves, Europ. J. Mech. B Fluids, 27 (2007), 96-109. doi: 10.1016/j.euromechflu.2007.04.004.

[10]

J. Ko and W. Strauss, Effect of vorticity on steady water waves, J. Fluid Mech., 608 (2008), 197-215. doi: 10.1017/S0022112008002371.

[11]

O. M. Philllips and M. L. Banner, Wave breaking in presence of wind drift and swell, J. Fluid Mech., 66 (1974), 625-640. doi: 10.1017/S0022112074000413.

[12]

J. Serrin, A symmetry theorem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304-318. doi: 10.1007/BF00250468.

[13]

W. Strauss, Steady water waves, Bull. Amer. Math. Soc. (N.S.), 47 (2010), 671-694.

[14]

S. Walsh, Stratified and steady periodic gravity waves, SIAM J. Math. Anal., 41 (2009), 1054-1105. doi: 10.1137/080721583.

[15]

S. Walsh, Steady periodic gravity waves with surface tension, preprint, arXiv:0911.1375.

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