Small movements and eigenoscillations of a system “an ideal fluid– barotropic gas”.

The paper is devoted to investigation of the problem on small movements and eigenoscillations of a system that consists of an ideal incompressible fluid and barotropic gas, and is situated in bounded vessel.
At the first part we use an operator approach for an investigation of the problem on small oscillations of the system. By this way the initial-boundary value problem is transformed to Cauchy problem for differential-operator equation of the second order in some Hilbert space. The operator coefficients of this equation are operator of kinetic energy (positive and compact) and operator of potential energy (bounded from below self-adjoint one with discrete spectrum). At the bounded statically stable equilibrium state the potential energy operator is positive definite.
On the base of the properties of these operators we prove the theorems on correct solvability as Cauchy problem for differential-operator equation as initial-boundary value problem. We prove also the theorem on instability of considered system in the case when an minimal eigenvalue of the potential energy operator is negative.
For eigenoscillations problem we prove that the spectrum of this problem is discrete with a limit point at infinity. For computing of the eigenvalues (squared oscillation freequences) we formulate the variational principle on the base of Ritz approach.
At the second part of our paper we consider more detailed the special case of the oscillations problem when the vessel is cylindrical with arbitrary cross section and an equilibrium dividing surface between fluid and gas is horizontal. We receive the characteristic equation of the problem and prove that solutions of the equation are asymptotically divided into two sets. For the first set there are corresponded gravitational-capillary waves in the system, and for the second one — acoustic waves in a gas.

Keywords: ideal fluid, incompressible fluid, barotropic gas, low gravity, equilibrium state, eigenoscillations, eigenvalue, operator approach, Hilbert space, initial-boundary value problem, spectral problem, solvability, strong solution, instability.