Hostname: page-component-7d8f8d645b-xs5cw Total loading time: 0 Render date: 2023-05-28T04:25:03.747Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Non-existence of three-dimensional travelling water waves with constant non-zero vorticity

Published online by Cambridge University Press:  01 April 2014

E. Wahlén*
Centre for Mathematical Sciences, Lund University, PO Box 118, 22100 Lund, Sweden
Email address for correspondence:


We prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves under a certain condition on the pressure at the surface, which is satisfied by sufficiently small waves. The proof relies on unique continuation arguments and Liouville-type results for elliptic equations.

© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Axler, S., Bourdon, P. & Ramey, W. 2001 Harmonic Function Theory, 2nd edn. Graduate Texts in Mathematics, vol.137, Springer.Google Scholar
Constantin, A. 2011a Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81. SIAM.Google Scholar
Constantin, A. 2011b Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train. Eur. J. Mech. (B/Fluids) 30 (1), 1216.CrossRefGoogle Scholar
Constantin, A. & Kartashova, E. 2009 Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves. Europhys. Lett. 86 (2), 29001; (6 pages).CrossRefGoogle Scholar
Constantin, A. & Strauss, W. 2004 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Maths 57 (4), 481527.CrossRefGoogle Scholar
Constantin, A. & Varvaruca, E. 2011 Steady periodic water waves with constant vorticity: regularity and local bifurcation. Arch. Rat. Mech. Anal. 199 (1), 3367.CrossRefGoogle Scholar
Craig, W. 2002 Non-existence of solitary water waves in three dimensions. In Recent Developments in the Mathematical Theory of Water Waves, Oberwolfach, Germany 2001. Phil. Trans. R. Soc. Lond. 360 (1799), 2127–2135.Google Scholar
Craig, W. & Nicholls, D. P. 2000 Travelling two and three dimensional capillary gravity water waves. SIAM J. Math. Anal. 32 (2), 323359.CrossRefGoogle Scholar
Dubreil-Jacotin, M. L. 1934 Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finite. J. Math. Pures Appl. 13, 217291.Google Scholar
Gerstner, F. 1809 Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 32, 412445.CrossRefGoogle Scholar
Groves, M. D. 2007 Three-dimensional travelling gravity–capillary water waves. GAMM-Mitt. 30 (1), 843.CrossRefGoogle Scholar
Groves, M. D. & Mielke, A. 2001 A spatial dynamics approach to three-dimensional gravity–capillary steady water waves. Proc. R. Soc. Edin. A 131 (1), 83136.CrossRefGoogle Scholar
Iooss, G. & Plotnikov, P. I. 2009 Small divisor problem in the theory of three-dimensional water gravity waves. Mem. Am. Math. Soc. 200 (940), viii+128.Google Scholar
Iooss, G. & Plotnikov, P. 2011 Asymmetrical three-dimensional travelling gravity waves. Arch. Rat. Mech. Anal. 200 (3), 789880.CrossRefGoogle Scholar
Johnson, R. S. 1997 A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press.CrossRefGoogle Scholar
Krylov, N. V. 1996 Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12. American Mathematical Society.Google Scholar
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9117.CrossRefGoogle Scholar
Reeder, J. & Shinbrot, M. 1981 Three-dimensional, nonlinear wave interaction in water of constant depth. Nonlinear Anal. 5 (3), 303323.CrossRefGoogle Scholar
Rudin, W. 1987 Real and Complex Analysis. 3rd edn. McGraw-Hill.Google Scholar
Stuhlmeier, R. 2012 On constant vorticity flows beneath two-dimensional surface solitary waves. J. Nonlinear Math. Phys. 19 (suppl. 1), 1240004; (9 pages).CrossRefGoogle Scholar