Normal oscillations of ideal stratified fluid with a free surface completely covered with the elastic 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 gravitational 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 the elastic ice.

Let us consider the basic case of stable stratification of the fluid on density:
\begin{equation*} \begin{split} &0<N_{min}^{2} \leq N^{2}(x_3) \leq N_{max}^{2} =: N_0^2 < \infty, \\ &N^2(x_3) := - \frac{ g\rho_0'(x_3) }{ \rho_0(x_3)},\quad\rho_0(0)>0, \end{split} \end{equation*}
where $N^{2}(x_3)$ is square frequency of buoyancy.

The initial boundary value problem is reduced to a Cauchy problem
\begin{equation*} \begin{split} &\mathcal A \frac{d^2x}{dt^2} + \mathcal C x = f(t), \quad x(0)=x^0, \quad x^{'}(0)=x^1, \\ &0<< \mathcal A= \mathcal A^{*} \in \mathcal L(\mathcal H), \quad \mathcal C = \mathcal C^{*} \geq 0. \end{split} \end{equation*}
in some Hilbert space $\mathcal H$.

The spectrum of normal oscillations, basic properties of eigenfunctions and other questions are studied.
Let us consider the basic case of stable stratification of the fluid on density:
\begin{equation*} \begin{split} &0<N_{min}^{2} \leq N^{2}(x_3) \leq N_{max}^{2} =: N_0^2 < \infty, \\ &N^2(x_3) := - \frac{ g\rho_0'(x_3) }{ \rho_0(x_3)},\quad\rho_0(0)>0, \end{split} \end{equation*}
where $N^{2}(x_3)$ is square frequency of buoyancy.
The initial boundary value problem is reduced to a Cauchy problem
\begin{equation*} \begin{split} &\mathcal A \frac{d^2x}{dt^2} + \mathcal C x = f(t), \quad x(0)=x^0, \quad x^{'}(0)=x^1, \\ &0<< \mathcal A= \mathcal A^{*} \in \mathcal L(\mathcal H), \quad \mathcal C = \mathcal C^{*} \geq 0. \end{split} \end{equation*}
in some Hilbert space $\mathcal H$.

The spectrum of normal oscillations, basic properties of eigenfunctions and other questions are studied.
Let us consider the basic case of stable stratification of the fluid on density:
\begin{equation*} \begin{split} &0<N_{min}^{2} \leq N^{2}(x_3) \leq N_{max}^{2} =: N_0^2 < \infty, \\ &N^2(x_3) := - \frac{ g\rho_0'(x_3) }{ \rho_0(x_3)},\quad\rho_0(0)>0, \end{split} \end{equation*}
where $N^{2}(x_3)$ is square frequency of buoyancy.
The initial boundary value problem is reduced to a Cauchy problem
\begin{equation*} \begin{split} &\mathcal A \frac{d^2x}{dt^2} + \mathcal C x = f(t), \quad x(0)=x^0, \quad x^{'}(0)=x^1, \\ &0<< \mathcal A= \mathcal A^{*} \in \mathcal L(\mathcal H), \quad \mathcal C = \mathcal C^{*} \geq 0. \end{split} \end{equation*}
in some Hilbert space $\mathcal H$.
The spectrum of normal oscillations, basic properties of eigenfunctions and other questions are studied.