zbMATH — the first resource for mathematics

Initial-boundary value problem for systems of equations describing wave motion in a fluid layer on an elastic base. (English. Russian original) Zbl 0781.76010
Math. Notes 52, No. 1, 748-750 (1992); translation from Mat. Zametki 52, No. 1, 150-153 (1992).
The title initial-boundary value problem is \(\nabla^ 2\psi= 0\), \(-H< z< 0\), \(\nabla^ 2 U+ (c^ 2- 1)\nabla\text{ div } U= \mu^ 2 U_{tt}\), \(z<- H\), \(\psi_{tt}+ \psi_ z= 0\), \(z= 0\), where \(\psi\) is the potential of the moving liquid \((-H\leq z\leq 0)\) and \(U\) is the displacement vector in the elastic half-space \(z\leq -H\). A sketch of the proof of the following statement is given: the system in question can be reduced to the form \(\partial^ 2 W/\partial t^ 2={\mathcal L}W\), where \(\mathcal L\) is a certain essentially self-adjoint operator generated by an explicitly written linear system of equations. The spectrum of \(\widehat{\mathcal L}\) consists of discrete and continuous parts, and the corresponding eigenfunctions form a complete orthonormal system with respect to a certain scalar product.
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] G. S. Pod”yapol’skii, Methods for Computing the Appearance and Propagation of Tsunamis [in Russian], Nauka, Moscow (1978).
[2] A. A. Shkalikov, Trudy Seminara im. I. G. Petrovskogo, No. 9, 190-230 (1983).
[3] N. V. Zvolinskii, Fiz. Zemli, No. 3, 3-14 (1986).
[4] D. G. Vasil’ev and V. B. Lidskii, Funkt. Anal., No. 3 (1981).
[5] S. Yu. Dobrokhotov and O. L. Tolstova, Mat. Zametki,47, No. 5, 148-151 (1990).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.