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The floating-body problem: an integro-differential equation without irregular frequencies


Author: N. Kuznetsov
Original publication: Algebra i Analiz, tom 31 (2019), nomer 3.
Journal: St. Petersburg Math. J. 31 (2020), 521-531
MSC (2010): Primary 31B10; Secondary 76B15, 35Q35
DOI: https://doi.org/10.1090/spmj/1612
Published electronically: April 30, 2020
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Abstract | References | Similar Articles | Additional Information

Abstract: The linear boundary value problem under consideration describes time-harmonic motion of water in a horizontal three-dimensional layer of constant depth in the presence of an obstacle adjacent to the upper side of the layer (floating body). This problem for a complex-valued harmonic function involves mixed boundary conditions and a radiation condition at infinity. Under rather general geometric assumptions the existence of a unique solution is proved for all values of the problem's nonnegative parameter related to the frequency of oscillations. The proof is based on the representation of a solution as a sum of simple- and double-layer potentials with densities distributed over the obstacle's surface, thus reducing the problem to an indefinite integro-differential equation. The latter is shown to be soluble for all continuous right-hand side terms, for which purpose S. G. Krein's theorem about indefinite equations is used.


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  • [1] David Colton and Rainer Kress, Integral equation methods in scattering theory, Classics in Applied Mathematics, vol. 72, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. Reprint of the 1983 original [ MR0700400]. MR 3397293
  • [2] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [3] N. M. Günter, Potential theory and its applications to basic problems of mathematical physics, Translated from the Russian by John R. Schulenberger, Frederick Ungar Publishing Co., New York, 1967. MR 0222316
  • [4] Fritz John, On the motion of floating bodies. II. Simple harmonic motions, Comm. Pure Appl. Math. 3 (1950), 45–101. MR 37118, https://doi.org/10.1002/cpa.3160030106
  • [5] R. E. Kleinman, On the mathematical theory of motion of floating bodies - an update, D. W. Taylor Naval Ship Res. & Devel. Center, Report 82/074, 1982.
  • [6] S. G. Kreĭn, On an indeterminate equation in Hilbert space and its application in potential theory, Usphei Matem. Nauk (N.S.) 9 (1954), no. 3(61), 149–153 (Russian). MR 0064307
  • [7] S. G. Kreĭn, Linear equations in Banach spaces, Birkhäuser, Boston, Mass., 1982. Translated from the Russian by A. Iacob; With an introduction by I. Gohberg. MR 684836
  • [8] Rainer Kress, Linear integral equations, 3rd ed., Applied Mathematical Sciences, vol. 82, Springer, New York, 2014. MR 3184286
  • [9] N. G. Kuznetsov, Steady waves on the surface of a water layer of variable depth with immersed floating bodies, Regular Asymptotic Algorithms in Mechanics, Nauka, Novosibirsk, 1989, pp. 200-261; 268-270. (Russian)
  • [10] N. G. Kuznetsov, Integral equations for a problem on steady waves induced by a floating body, Mat. Zametki 50 (1991), no. 4, 75–83, 159 (Russian); English transl., Math. Notes 50 (1991), no. 3-4, 1036–1042 (1992). MR 1162914, https://doi.org/10.1007/BF01137734
  • [11] N. Kuznetsov, V. Maz’ya, and B. Vainberg, Linear water waves, Cambridge University Press, Cambridge, 2002. A mathematical approach. MR 1925354
  • [12] A. M. Liapounoff, Sur le potentiel de la double couche, Comm. Soc. Math. Kharkow. Sér. 2 6 (1899), 129-138.
  • [13] V. V. Luk′yanov and A. I. Nazarov, Solution of the Venttsel′problem for the Laplace and the Helmholtz equations by means of iterated potentials, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 250 (1998), no. Mat. Vopr. Teor. Rasprostr. Voln. 27, 203–218, 337–338 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 102 (2000), no. 4, 4265–4274. MR 1701867, https://doi.org/10.1007/BF02673857
  • [14] M. McIver, An example of non-uniqueness in the two-dimensional linear water wave problem, J. Fluid Mech. 315 (1996), 257–266. MR 1403701, https://doi.org/10.1017/S0022112096002418
  • [15] P. McIver and M. McIver, Trapped modes in an axisymmetric water-wave problem, Quart. J. Mech. Appl. Math. 50 (1997), no. 2, 165–178. MR 1451065, https://doi.org/10.1093/qjmam/50.2.165
  • [16] Vladimir Gilelevič Maz′ja and Jürgen Rossmann, Über die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten, Math. Nachr. 138 (1988), 27–53 (German). MR 975198, https://doi.org/10.1002/mana.19881380103
  • [17] S. G. Mikhlin, Mathematical physics, an advanced course, With appendices by V. M. Babič, V. G. Maz′ja and I. Ja. Bakel′man. Translated from the Russian. North-Holland Series in Applied Mathematics and Mechanics, Vol. 11, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. MR 0286325
  • [18] V. A. Steklov, Osnovnye zadachi matematicheskoĭ fiziki, 2nd ed., “Nauka”, Moscow, 1983 (Russian). Edited and with a preface by V. S. Vladimirov. MR 721642
  • [19] F. Ursell, Irregular frequencies and the motion of floating bodies, J. Fluid Mech. 105 (1981), 143–156. MR 617683, https://doi.org/10.1017/S0022112081003145
  • [20] Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, https://doi.org/10.1016/0022-1236(84)90066-1

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Additional Information

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, 199178 St. Petersburg, Russian Federation
Email: nikolay.g.kuznetsov@gmail.com

DOI: https://doi.org/10.1090/spmj/1612
Keywords: Potential representations, integral operators, integro-differential equation
Received by editor(s): August 20, 2018
Published electronically: April 30, 2020
Dedicated: In memoriam of my mentor S. G. Mikhlin
Article copyright: © Copyright 2020 American Mathematical Society