Maz’ya’s works in the linear theory of water waves

  • N. G. Kuznetsov
  • B. R. Vainberg
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 109)


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.


Free Surface Asymptotic Expansion Linear Theory Water Wave Uniqueness Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Publications of Vladimir Maz’ya

  1. 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.Google Scholar
  2. 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.Google Scholar
  3. 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.Google Scholar
  4. 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.zbMATHGoogle Scholar
  5. 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).MathSciNetGoogle Scholar
  6. 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.CrossRefGoogle Scholar
  7. 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).Google Scholar
  8. 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).Google Scholar
  9. 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.MathSciNetzbMATHGoogle Scholar
  10. Kuznetsov, N.G. & Maz’ya, V.G. 1989 On unique solvability of the plane Neumann-Kelvin problem. Math. USSR Shorn. 63, 425–446.MathSciNetCrossRefGoogle Scholar
  11. 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.Google Scholar
  12. 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.Google Scholar
  13. Maz’ya, V.G. & Vainberg, B.R. 1993 On ship waves. Wave Motion 18, 31–50.MathSciNetCrossRefGoogle Scholar
  14. Livshits, M. & Maz’ya, V. 1997 Solvability of the two-dimensional Kelvin-Neumann problem for a submerged circular cylinder. Applicable Analysis 64, 1–5.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 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.MathSciNetzbMATHCrossRefGoogle Scholar


  1. Lamb, H. 1932 Hydrodynamics. Cambridge: Camb. Univ. Press.zbMATHGoogle Scholar
  2. 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)).Google Scholar
  3. 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)).Google Scholar
  4. 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)).Google Scholar
  5. John, F. 1950 On the motion of floating bodies. II. Comm. Pure Appl. Math. 3, 45–101.MathSciNetCrossRefGoogle Scholar
  6. 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.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 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.zbMATHCrossRefGoogle Scholar
  9. Stoker, J.J. 1957 Water Waves. The Mathematical Theory with Applications. New York: Intersci. Publ.zbMATHGoogle Scholar
  10. 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.Google Scholar
  11. 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.Google Scholar
  12. 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.Google Scholar
  13. Angell, T.S. & Kleinman, R.E. 1984 A Galerkin procedure for optimization in radiation problems. SIAM J. Appl. Math. 44, 1246–1257.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 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.MathSciNetCrossRefGoogle Scholar
  15. Lahalle, D. 1984 Calcul des efforts sur un profil portant d’hydroptere par couplage éléments finis — représentation intégrale. ENSTA Rapport de Recherche 187.Google Scholar
  16. Angell, T.S., Hsiao, G.C. & Kleinman, R.E. 1986 An optimal design problem for submerged bodies. Math. Meth. Appl. Sci. 8, 50–76.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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.Google Scholar
  18. 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).Google Scholar
  19. Weck, N. 1990 On a boundary value problem in the theory of linear water-waves. Math. Meth. Appl. Sci. 12, 393–404.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Angell, T.S., Kleinman, R.E., 1991 A constructive method for shape optimization: a problem in hydrodynamics. IMA J. Appl. Math. 47, 265–281.MathSciNetzbMATHCrossRefGoogle Scholar
  21. Kuznetsov, N.G. 1991 Uniqueness of a solution of a linear problem for stationary oscillations of a liquid. Diff. Equat. 27, 187–194.zbMATHGoogle Scholar
  22. 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.MathSciNetCrossRefGoogle Scholar
  23. Ursell, F. 1992 Some unsolved and unfinished problems in the theory of waves. Wave Asymptotics. Cambridge: Camb. Univ. Press.Google Scholar
  24. Kuznetsov, N.G. 1993a Asymptotic analysis of wave resistance of a submerged body moving with an oscillating velocity. J. Ship Res. 37, 119–125.Google Scholar
  25. 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.Google Scholar
  26. 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.Google Scholar
  27. McIver, M. 1996 An example of non-uniqueness in the two-dimensional linear water wave problem. J. Fluid Mech. 315, 257–266.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 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.Google Scholar
  29. 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.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 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.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • N. G. Kuznetsov
    • 1
  • B. R. Vainberg
    • 2
  1. 1.Laboratory for Mathematical Modelling of Wave PhenomenaInstitute of Problems in Mechanical Engineering, Russian Academy of SciencesSt. PetersburgRussian Federation
  2. 2.University of North Carolina at CharlotteCharlotteUSA

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