Small motions of ideal stratified liquid with a free surface totally covered by a crumbled ice

Let a rigid immovable vessel be partially filled with an ideal incompressible stratified fluid. We assume that in an equilibrium state the density of a fluid is a function of the vertical variable $x_3,$ i.e., $\rho_0=\rho_0(x_3).$ In this case the gravity field with constant acceleration $\vec g=-g\vec e_3$ acts on the fluid, here $g>0$ and $\vec e_3$ is unit vector of the vertical axis $Ox_3,$ which is directed opposite to $\vec g.$ Let $\Omega$ be the domain filled with a fluid in equilibrium state, $S$ be rigid wall of the vessel adherent to the fluid, $\Gamma$ be a free surface completely covered with a crumbled ice. As the crumbled ice we mean that on the free surface, weighty particles of some substance float, and these particles do not interact or the interaction is negligible as the free surface oscillates. We should note that in foreign publications, such fluids are frequently called liquids with inertial free surfaces. 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 second-order equation in Hilbert space. After a detailed study of the properties of the operator coefficients corresponding to the resulting system of equations, we prove a theorem on the strong solvability of the Cauchy problem obtained on a finite time interval. On this base, we find sufficient conditions for the existence of a strong (with respect to time variable) solution to the initial-boundary value problem describing the evolution of the hydrosystem.