Abstract
The paper surveys results of V. Maz’ya in the linear theory of water waves. All main topics of his work in this field are considered. At first, we describe Maz’ya’s achievements concerning the tough question of the unique solvability of two steady-state problems, which are: (1) the problem of time-harmonic waves in a layer of variable depth, and above a totally submerged body; (2) the problem of wave patterns due to the uniform forward motion of a body in water of constant depth. The review ends with a description of asymptotic expansions for unsteady waves arising from brief and high-frequency disturbances. A complete list of Maz’ya’s publications on water waves is given.
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References
Publications of Vladimir Maz’ya
Vainberg, B.R. & Maz’ja, V.G. 1972 On some stationary problems in the linear theory of surface waves. Soviet Physics Dokl. 17, 640–643.
Vainberg, B.R. & Maz’ja, V.G. 1973a On the plane problem of the motion of a body immersed in a fluid. Trans. Moscow Math. Soc. 28, 33–55.
Vainberg, B.R. & Maz’ja, V.G. 1973b On the problem of the steady state oscillations of a fluid layer of variable depth. Trans. Moscow Math. Soc. 28, 56–73.
Kuznetsov, N.G. & Maz’ya, V.G. 1974 Problem concerning steady-state oscillations of a layer of fluid in the presence of an obstacle. Soviet Physics Dokl. 19, 341–343.
Maz’ya, V.G. 1977 On the steady problem of small oscillations of a fluid in the presence of a submerged body. Proc. Sobolev’s Semin. No. 2, 57–79. Novosibirsk: Inst. of Maths, Siberian Branch, Acad. Sci. USSR (in Russian).
Maz’ja, V.G. 1978 Solvability of the problem on the oscillations of a fluid containing a submerged body. J. Soviet Math. 10, 86–89.
Kuznetsov, N.G. & Maz’ya, V.G. 1985 Asymptotic expansions for transient surface waves due to short-period oscillating disturbances. Proc. Leningrad Ship-build. Inst. / Math. Modelling and Automated Design in Shipbuilding, 57–64 (in Russian).
KuzNetsov, N.G. & Maz’ya, V.G. 1987 Asymptotic expansions for surface waves caused by rapidly oscillating or accelerating disturbances. Asymptotic methods / Problems and Models in Mechanics. Novosibirsk: ‘nauka’, pp. 136–175 (in Russian).
Kuznetsov, N.G. & Maz’ya, V.G. 1988 Unique solvability of a plane stationary problem related to the motion of a solid body submerged in a liquid. Diff. Equat. 24, 1291–1301.
Kuznetsov, N.G. & Maz’ya, V.G. 1989 On unique solvability of the plane Neumann-Kelvin problem. Math. USSR Shorn. 63, 425–446.
Kuznetsov, N.G. & Maz’ya, V.G. 1992 On a well-posed formulation of the two-dimensional Neumann-Kelvin problem for a surface-piercing body. Preprint LiTH-MAT-R-92-42, Dept. of Maths, University of Linköping, 34 p.
Maz’ya, V. & Vainberg, B. 1992 On uniqueness and asymptotic behavior of solutions of the Neumann-Kelvin problem. Proc. of the 7th Int. Workshop on Water Waves & Floating Bodies, France. Ed. R. Cointe, pp. 177–181.
Maz’ya, V.G. & Vainberg, B.R. 1993 On ship waves. Wave Motion 18, 31–50.
Livshits, M. & Maz’ya, V. 1997 Solvability of the two-dimensional Kelvin-Neumann problem for a submerged circular cylinder. Applicable Analysis 64, 1–5.
Kuznetsov, N.G. & Maz’ya, V.G. 1997 Asymptotic analysis of surface waves due to high-frequency disturbances. Rend. Mat. Acc. Lincei, Ser. 9, 8, 5–29.
Other_Works
Lamb, H. 1932 Hydrodynamics. Cambridge: Camb. Univ. Press.
Kochin, N.E. 1937 On the wave resistance and lift of bodies submerged in a fluid. Proc. Con, on the Wave Resistance Theory. Moscow: TsAGI, pp. 65–134. (In Russian; English transi, in SN AME Tech. & Res. Bull. 1-8 (1951)).
Kochin, N.E. 1939 The two-dimensional problem of steady oscillations of bodies under the free surface of a heavy incompressible fluid. Acad. Sci. USSR, Izvestia OTN, No. 4, 37–62. (In Russian; English transl. in SNAME Tech. & Res. Bull. 1-10 (1952)).
Kochin, N.E. 1940 The theory of waves generated by oscillations of a body under the free surface of a heavy incompressible fluid. Trans. Moscow Univ. 46, 85–106. (In Russian; English transl. in SNAME Tech. & Res. Bull. 1-10(1952)).
John, F. 1950 On the motion of floating bodies. II. Comm. Pure Appl. Math. 3, 45–101.
Ursell, F. 1950 Surface waves on deep water in the presence of a submerged circular cylinder. I, II. Proc. Camb. Phil. Soc. 46, 141–152, 153-158.
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.
Jones, D.S. 1953 The eigenvalue of ∇2u+λu = 0 when the boundary conditions are given on semiinfinite domains, Proc. Camb. Phil. Soc., 49, 668–684.
Stoker, J.J. 1957 Water Waves. The Mathematical Theory with Applications. New York: Intersci. Publ.
Gohberg, I. & Krein, M.G. 1969 Introduction to the Theory of Linear Non-self-adjoint Operators in Hilbert Space. Transl. Math. Mon. 18. Providence, RI: Amer. Math. Soc.
Ursell, F. 1981 Mathematical notes on the two-dimensional Kelvin-Neumann problem. Proceedings of the 13th Symposium on Naval Hydrodynamics. Tokyo: Shipbuilding Research Association of Japan, pp. 245–251.
Suzuki, K. 1982 Numerical studies of the Neumann-Kelvin problem for a two-dimensional semisubmerged body. Proceedings of the 3d International Conference on Numerical Ship Hydrodynamics. Paris: Bassin d’essais des Carènes, pp. 83–95.
Angell, T.S. & Kleinman, R.E. 1984 A Galerkin procedure for optimization in radiation problems. SIAM J. Appl. Math. 44, 1246–1257.
Hulme, A. 1984 Some applications of Maz’ja’s uniqueness theorem to a class of linear water wave problems. Math. Proc. Camb. Phil. Soc. 95,511–519.
Lahalle, D. 1984 Calcul des efforts sur un profil portant d’hydroptere par couplage éléments finis — représentation intégrale. ENSTA Rapport de Recherche 187.
Angell, T.S., Hsiao, G.C. & Kleinman, R.E. 1986 An optimal design problem for submerged bodies. Math. Meth. Appl. Sci. 8, 50–76.
Angell, T.S. & Kleinman, R.E. 1987 On a domain optimization problem in hydrodynamics. Optimal Control of Partial Differential Equations. II. Basel et al.: Birkhäuser, pp. 9–27.
Kuznetsov, N.G. 1989 Steady waves on the surface of fluid having variable depth and containing floating bodies. Part 4 in: N.G. Kuznetsov, Yu.F. Orlov, V.B. Cherepennikov, R Yu. Shlaustas, Regular Asymptotic Algorithms in Mechanics. Novosibirsk: ‘nauka’, pp. 200–270 (in Russian).
Weck, N. 1990 On a boundary value problem in the theory of linear water-waves. Math. Meth. Appl. Sci. 12, 393–404.
Angell, T.S., Kleinman, R.E., 1991 A constructive method for shape optimization: a problem in hydrodynamics. IMA J. Appl. Math. 47, 265–281.
Kuznetsov, N.G. 1991 Uniqueness of a solution of a linear problem for stationary oscillations of a liquid. Diff. Equat. 27, 187–194.
Kuznetsov, N.G. 1992 The lower bound of the eigenfrequencies of plane oscillations of a fluid in a channel. J. Appl. Math. Mech. 56, 293–297.
Ursell, F. 1992 Some unsolved and unfinished problems in the theory of waves. Wave Asymptotics. Cambridge: Camb. Univ. Press.
Kuznetsov, N.G. 1993a Asymptotic analysis of wave resistance of a submerged body moving with an oscillating velocity. J. Ship Res. 37, 119–125.
Kuznetsov, N.G. 1993b The Maz’ya identity and lower estimates of eigenfrequencies of steady-state oscillations of a liquid in a channel. Russian Math. Surveys 48(4), 222.
KuzNetsov, N.G. & Simon, M.J. 1995 On uniqueness in the two-dimensional water-wave problem for surface-piercing bodies in fluid of finite depth. Appl. Math. Rep. 95/4. University of Manchester.
McIver, M. 1996 An example of non-uniqueness in the two-dimensional linear water wave problem. J. Fluid Mech. 315, 257–266.
Evans, D.V. & Kuznetsov, N.G. 1997 Trapped modes. In: Gravity Waves in Water of Finite Depth (ed. J.N. Hant), pp. 127–168, Comp. Mech. Int., Southampton.
Kuznetsov, N. & Motygin, O. 1997 On waveless statement of the two-dimensional Neumann-Kelvin problem for a surface-piercing body. IMA J. Appl. Math. 59, 25–42.
Kuznetsov, N. & Motygin, O. 1999 On the resistanceless statement of the two-dimensional Neumann-Kelvin problem for a surface-piercing tandem. IMA J. Appl. Math. 62, 1–18.
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Kuznetsov, N.G., Vainberg, B.R. (1999). Maz’ya’s works in the linear theory of water waves. In: Rossmann, J., Takáč, P., Wildenhain, G. (eds) The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol 109. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8675-8_3
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