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Waves in a fluid over an elastic bottom. The existence theorem and exact solutions. (English. Russian original) Zbl 0830.35099
Math. Notes 54, No. 6, 1208-1222 (1993); translation from Mat. Zametki 54, No. 6, 33-55 (1993).
We study a linear model that describes waves in a fluid over an elastic bottom. We state a mathematically precise Cauchy problem for this model. We first derive existence and uniqueness theorems, obtaining some a priori estimates for the solution, and then obtain some exact and approximate solutions (quasi-classical asymptotics).
In addition, by the physical formulation of the problem we are interested in the case where the reservoir bottom is uneven, which produces variable coefficients. We also study the Cauchy problem for sufficiently large times.

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
74B05 Classical linear elasticity
Full Text: DOI
[1] S. Yu. Dobrokhotov and P. N. Zhevandrov, ?Computation of ocean waves over a variable bottom using Maslov’s method,? The Study of Tsunamis, No. 1, 73-79 (1986).
[2] P. N. Zhevandrov, ?The Cauchy-Poisson problem for gravity-capillary waves in water of variable depth,? Zh. Vychisl. Matem. Mat. Fiz.,27, No. 12, 1834-1844 (1987).
[3] P. C. Sabatier, ?On water waves produced by ground motions,? J. Fluid Mech.,126, 27-58 (1983). · Zbl 0528.76020 · doi:10.1017/S0022112083000038
[4] G. S. Pod’yapol’skii, ?Generation of tsunamis by earthquakes,? Methods for Computing the Appearance and Propagation of Tsunamis [in Russian], Nauka, Moscow (1978), pp. 30-87.
[5] N. V. Zvolinksii, I. S. Nikitin, and S. Ya. Sekerzh-Zen’kovich, ?Generation of tsunami waves and Rayleigh waves by a harmonic expansion center,? Izv. Akad. Nauk SSSR, Earth Physics, No. 2, 34-44 (1991).
[6] I. A. Molotkov and P. V. Kraukliss, ?Mixed surface waves on the boundary between an elastic medium and fluid,? Izv. Akad. Nauk SSSR, Earth Physics, No. 8 (1971).
[7] A. G. Aslanyan, D. G. Vasil’ev, and V. B. Lidskii, ?The frequency of free oscillations of a thin envelope interacting with a fluid,? Funkts. Anal.,15, No. 3, 1-9 (1981).
[8] R. M. Garipov, ?On the theory of gravity waves: the theorem of existence and uniqueness,? Arch. Rat. Mech. Anal.,24, No. 5, 352-362 (1967). · Zbl 0149.45603 · doi:10.1007/BF00253152
[9] M. S. Agranovich, ?Boundary problems for systems with a parameter,? Mat. Sbornik,84(126), No. 1, 27-65 (1971). · Zbl 0207.10801
[10] R. Sakamoto, ?Mixed problems for hyperbolic equations. I, II,? Matematika,16, No. 1, 62-100 (1972).
[11] H. O. Kreiss, ?Initial boundary value problems for hyperbolic systems,? Comm. Pure Appl. Math.,23, 277-298 (1970). · Zbl 0193.06902 · doi:10.1002/cpa.3160230304
[12] Yu. M. Berezanskii, Expansion of Self-Adjoint Operators into Eigenfunctions [in Russian], Naukova Dumka, Kiev (1965).
[13] S. Misohata, Theory of Partial Differential Equations [Russian translation], Mir, Moscow (1977).
[14] H. Duveau and J.-L. Lyons, The Inequality in Mechanics and Physics [Russian translation], Nauka, Moscow (1980).
[15] J.-L. Lyons and E. Magenes, Nonhomogeneous Boundary Value Problems and Their Applications [Russian translation], Mir, Moscow (1971).
[16] P. N. Zhevandrov, ?Asymptotic solutions of problems of non-stabilized waves on a fluid surface in a reservoir with an uneven bottom,? Candidate’s Dissertation, Physical and Mathematical Sciences, Moscow (1985).
[17] M. V. Fedoryuk, ?Singularities of kernels of Fourier integral operators and the asymptotics for the solution to the mixed problem,? Usp. Mat. Nauk,32, No. 6, 67-115 (1977).
[18] S. Yu. Dobrokhotov and O. L. Tolstova, ?On the application of conservation laws in asymptotic problems for equations with an operator-valued symbol,? Mat. Zametki,47, No. 5, 148-151 (1990). · Zbl 0731.35062
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