Localization Principle and an Estimate of the Rate of Damping of Oscillations in a Viscous Compressible Stratified Liquid

We consider the initial boundary-value problem describing small acoustic-gravitational oscillations of a viscous stratified liquid in the linear approximation. A distinctive feature of the present paper is the rejection of the Boussinesq assumption concerning the stationary density, and this leads to the study of a hydrodynamic system with variable coefficients. Using the localization principle, we obtain an estimate for the rate of stabilization of the solution to the problem as $t\to\infty$. The localization principle is based on the relationship between the rate of decrease of the solution to the problem and the geometry of the domain of loss of analyticity of the Laplace image ($L_{t\to\gamma}$) of the solution. The localization principle is based on the use of a priori estimates for the image of the solution. The unique solvability of both the original problem and that in terms of images is proved along the same lines.