Open AccessAsymptotics of the Spectrum of Variational Problems Arising in the Theory of Fluid Oscillations
Author(s) -
T. A. Suslina
Publication year - 2021
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2021-67-2-363-407
Subject(s) - mathematics , mathematical analysis , spectrum (functional analysis) , boundary value problem , perfect fluid , variational principle , eigenvalues and eigenvectors , calculus of variations , domain (mathematical analysis) , well posed problem , physics , mathematical physics , quantum mechanics
This work is a survey of results on the asymptotics of the spectrum of variational problems arising in the theory of small oscillations of a fluid in a vessel near the equilibrium position. The problems were posed by N. D. Kopachevsky in the late 1970s and cover various fluid models. The formulations of problems are given both in the form of boundary-value problems for eigenvalues in the domain R3, which is occupied by the fluid in the equilibrium state, and in the form of variational problems on the spectrum of the ratio of quadratic forms. The common features of all the problems under consideration are the presence of an elliptic constraint (the Laplace equation for an ideal fluid or a homogeneous Stokes system for a viscous fluid), as well as the occurrence of the spectral parameter in the boundary condition on the free (equilibrium) surface . The spectrum in the considered problems is discrete; the spectrum distribution functions have power-law asymptotics.