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On the Benjamin–Lighthill conjecture for water waves with vorticity

Published online by Cambridge University Press:  24 July 2017

V. Kozlov ,
N. Kuznetsov Open the ORCID record for N. Kuznetsov [Opens in a new window]  and
E. Lokharu
V. Kozlov
Affiliation:
Department of Mathematics, Linköping University, S-581 83 Linköping, Sweden
N. Kuznetsov*
Affiliation:
Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61, St Petersburg 199178, Russia
E. Lokharu
Affiliation:
Department of Mathematics, Linköping University, S-581 83 Linköping, Sweden
*
†Email address for correspondence: [email protected]
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Abstract

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 961 - 1001
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© 2017 Cambridge University Press 

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References

Agmon, S., Douglis, A. & Nirenberg, L. 1959 Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Maths 12, 623–727.CrossRefGoogle Scholar
Amick, C. J. & Toland, J. F. 1981 On periodic water-waves and their convergence to solitary waves in the long-wave limit. Phil. Trans. R. Soc. Lond. A 303, 633–669.Google Scholar
Benjamin, T. B. 1984 Impulse, flow force and variational principles. IMA J. Appl. Maths 32, 3–68.CrossRefGoogle Scholar
Benjamin, T. B. 1995 Verification of the Benjamin–Lighthill conjecture about steady water waves. J. Fluid Mech. 295, 337–356.Google Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448–460.Google Scholar
Cabre, X. 1995 On the Alexandroff–Bakelman–Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations. Commun. Pure Appl. Maths 48, 539–570.CrossRefGoogle Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
Constantin, A. & Strauss, W. 2004 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Maths 57, 481–527.CrossRefGoogle Scholar
Constantin, A. & Strauss, W. 2011 Periodic travelling gravity water waves with discontinuous vorticity. Arch. Rat. Mech. Anal. 202, 133–175.Google Scholar
Dubreil-Jacotin, M.-L. 1934 Sur la détermination rigoureuse des ondes permanentes periodiques d’ampleur finie. J. Math. Pures Appl. 13, 217–291.Google Scholar
Ehrnström, M., Escher, J. & Villari, G. 2012 Steady water waves with multiple critical layers: interior dynamics. J. Math. Fluid Mech. 14, 407–419.Google Scholar
Ehrnström, M., Escher, J. & Wahlén, E. 2011 Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43, 1436–1456.Google Scholar
Gilbarg, D. & Trudinger, N. S. 2001 Elliptic Partial Differential Equations of Second Order. Springer.Google Scholar
Groves, M. D. & Stylianou, A. 2014 On the Hamiltonian structure of the planar steady water-wave problem with vorticity. C. R. Acad. Sci. Paris I 352, 205–211.Google Scholar
Groves, M. D. & Wahlén, E. 2008 Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity. Physica D 237, 1530–1538.Google Scholar
Keady, G. & Norbury, J. 1978 Waves and conjugate streams with vorticity. Mathematika 25, 129–150.Google Scholar
Kozlov, V. & Kuznetsov, N. 2009 Fundamental bounds for arbitrary steady water waves. Math. Ann. 345, 643–655.CrossRefGoogle Scholar
Kozlov, V. & Kuznetsov, N. 2010 The Benjamin–Lighthill conjecture for near-critical values of Bernoulli’s constant. Arch. Rat. Mech. Anal. 197, 433–488.CrossRefGoogle Scholar
Kozlov, V. & Kuznetsov, N. 2011a The Benjamin–Lighthill conjecture for steady water waves (revisited). Arch. Rat. Mech. Anal. 201, 631–645.Google Scholar
Kozlov, V. & Kuznetsov, N. 2011b Steady free-surface vortical flows parallel to the horizontal bottom. Q. J. Mech. Appl. Maths 64, 371–399.Google Scholar
Kozlov, V. & Kuznetsov, N. 2013a No steady water waves of small amplitude are supported by a shear flow with a still free surface. J. Fluid Mech. 717, 523–534.Google Scholar
Kozlov, V. & Kuznetsov, N. 2013b Steady water waves with vorticity: spatial Hamiltonian structure. J. Fluid Mech. 733, R1.Google Scholar
Kozlov, V. & Kuznetsov, N. 2014 Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents. Arch. Rat. Mech. Anal. 214, 971–1018.Google Scholar
Kozlov, V., Kuznetsov, N. & Lokharu, E. 2015 On bounds and non-existence in the problem of steady waves with vorticity. J. Fluid Mech. 765, R1.CrossRefGoogle Scholar
Kozlov, V. & Maz’ya, V. 1999 Differential Equations with Operator Coefficients with Applications to Boundary Value Problems for Partial Differential Equations. Springer.Google Scholar
Kozlov, V. & Maz’ya, V. 2004 An asymptotic theory of higher-order operator differential equations with non-smooth nonlinearities. J. Funct. Anal. 217, 448–488.Google Scholar
Ladyzhenskaya, O. A. & Uraltseva, N. N. 1964 Linear and Quasilinear Elliptic Equations, Nauka (in Russian); English transl., Academic, 1968.Google Scholar
Longuet-Higgins, M. S. & Fenton, J. D. 1974 On the mass, momentum, energy and circulation of a solitary wave II. Proc. R. Soc. Lond. A 340, 471–493.Google Scholar
McLeod, J. B. 1984 The Froude number for solitary waves. Proc. R. Soc. Edin. A 97, 193–197.Google Scholar
Mielke, A. 1991 Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems, Lecture Notes in Mathematics, vol. 1489. Springer.Google Scholar
Ovsyannikov, L. V. 1980 Parameters of cnoidal waves. In Problems of Mathematics and Mechanics. M. A. Lavrentiev Memorial Volume, pp. 150–166. Nauka (in Russian).Google Scholar
Peregrine, D. H. 1976 Interaction of water waves and currents. Adv. Appl. Mech. 16, 9–117.CrossRefGoogle Scholar
Plotnikov, P. I. 1992 Non-uniqueness of solution of the problem of solitary waves and bifurcation of critical points of smooth functionals. Math. USSR Izvestia 38, 333–357.Google Scholar
Swan, C., Cummins, I. & James, R. 2001 An experimental study of two-dimensional surface water waves propagating in depth-varying currents. J. Fluid Mech. 428, 273–304.CrossRefGoogle Scholar
Wahlén, E. 2009 Steady water waves with a critical layer. J. Differ. Equ. 246, 2468–2483.CrossRefGoogle Scholar
Wheeler, M. H. 2015 The Froude number for solitary water waves with vorticity. J. Fluid Mech. 768, 91–112.Google Scholar