Operator approach to the problem on small movements of stratified fluids

Let immovable container be completely filled with system of three nonmixing heavy stratified incompressible fluids. The lower fluid (with respect to gravity) is viscous with coefficient of dynamical viscosity $\mu_1 = const > 0$, middle fluid is ideal, upper one is viscous with coefficient of dynamical viscosity $\mu_3 = const > 0$. We assume that in an equilibrium state the densities of a fluids is a function of the vertical variable $x_3$, i.e., $\rho_i = \rho_i(x_3)$ $(i = \overline{1, 3}).$ In this case the gravitational field with constant acceleration $\overrightarrow{g} = −g\overrightarrow{e}_3$ acts on the fluids, here $g > 0$ and $\overrightarrow{e}_3$ is unit vector of the vertical axis $Ox_3$, which is directed opposite to $\overrightarrow{g}.$
Let us consider the basic case of stable stratification of the fluids on densities:
$$
0 < N^2_{i, min} \le N^2_i(x_3) \le N^2_{i, max} =: N^2_{0, i} < \infty,
\quad
N^2_i(x_3) := -\frac{g\rho^\prime_i(x_3)}{\rho_i(x_3)},
\quad
(i = \overline{1, 3}).
$$
The problem on small oscillations is studied on the base of approach connected with application of so-called operator matrices theory with unbounded entries. 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
$$
\mathscr{R}\frac{dy}{dt} + \mathscr{A}y = f(t),
\quad
y(0) = y^0,
\quad
0 \ll \mathscr{R} = \mathscr{R}^\ast \in \mathscr{L}(\mathscr{H}),
\quad
Re\mathscr{A} \ge 0,
$$
in some Hilbert space $\mathscr{H}$. The theorem on strong solvability of initial boundary value problem is proved.

Keywords: stratification effect in viscous and ideal fluids, differential equation in Hilbert space, accretive operator, strong solution.