The problem of normal oscillations of a viscous rotating stratified fluid

Assume that a viscous stratified fluid partially fills an arbitrary container and, in the unperturbed state, uniformly rotates with angular velocity ${\vec w_0 = \omega_0 \vec e_3}$, where ${\vec e_3}$ is the axial vector of rotation axis ${Ox_3}$ (assume that ${\omega_0>0}$). In a state of relative equilibrium, the fluid occupies the region ${\Omega \in \Bbb R^3}$ bounded by solid wall $S$ and the equilibrium surface ${\Gamma}$. Consider small motions of a fluid near equilibrium state. The problem is studied on the base of an approach connected with application of so-called operator matrices theory. To this end, we introduce Hilbert spaces and some their subspaces, also auxiliary boundary value problems. The initial boundary value problem is reduced to the Cauchy problem for the differential first-order equation in Hilbert space. After a detailed study of the properties of the