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

Kuznetsov, N.

doi:10.1090/spmj/1612

The floating-body problem: an integro-differential equation without irregular frequencies
HTML articles powered by AMS MathViewer

by N. Kuznetsov

St. Petersburg Math. J. 31 (2020), 521-531

DOI: https://doi.org/10.1090/spmj/1612

Published electronically: April 30, 2020

PDF | Request permission

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.

References

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, DOI 10.1137/1.9781611973167.ch1
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, DOI 10.1007/978-3-642-61798-0
N. M. Günter, Potential theory and its applications to basic problems of mathematical physics, Frederick Ungar Publishing Co., New York, 1967. Translated from the Russian by John R. Schulenberger. MR 0222316
Fritz John, On the motion of floating bodies. II. Simple harmonic motions, Comm. Pure Appl. Math. 3 (1950), 45–101. MR 37118, DOI 10.1002/cpa.3160030106
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.
S. G. Kreĭn, On an indeterminate equation in Hilbert space and its application in potential theory, Uspehi Matem. Nauk (N.S.) 9 (1954), no. 3(61), 149–153 (Russian). MR 0064307
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
Rainer Kress, Linear integral equations, 3rd ed., Applied Mathematical Sciences, vol. 82, Springer, New York, 2014. MR 3184286, DOI 10.1007/978-1-4614-9593-2
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)
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, DOI 10.1007/BF01137734
N. Kuznetsov, V. Maz’ya, and B. Vainberg, Linear water waves, Cambridge University Press, Cambridge, 2002. A mathematical approach. MR 1925354, DOI 10.1017/CBO9780511546778
A. M. Liapounoff, Sur le potentiel de la double couche, Comm. Soc. Math. Kharkow. Sér. 2 6 (1899), 129–138.
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, DOI 10.1007/BF02673857
M. McIver, An example of non-uniqueness in the two-dimensional linear water wave problem, J. Fluid Mech. 315 (1996), 257–266. MR 1403701, DOI 10.1017/S0022112096002418
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, DOI 10.1093/qjmam/50.2.165
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, DOI 10.1002/mana.19881380103
S. G. Mikhlin, Mathematical physics, an advanced course, North-Holland Series in Applied Mathematics and Mechanics, Vol. 11, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1970. With appendices by V. M. Babič, V. G. Maz′ja and I. Ja. Bakel′man; Translated from the Russian. MR 0286325
V. A. Steklov, Osnovnye zadachi matematicheskoĭ fiziki, 2nd ed., “Nauka”, Moscow, 1983 (Russian). Edited and with a preface by V. S. Vladimirov. MR 721642
F. Ursell, Irregular frequencies and the motion of floating bodies, J. Fluid Mech. 105 (1981), 143–156. MR 617683, DOI 10.1017/S0022112081003145
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, DOI 10.1016/0022-1236(84)90066-1