Small motions of the system of two viscous stratification fluids.

Let immovable container be completely filled with system of two viscous stratified incompressible fluids. We assume that in an equilibrium state the densities of a fluids is a function of the vertical variable $x_{3}$, i.e., $\rho_{i}=\rho_{i} \left ( x_{3} \right )$ $\left ( i=1,2 \right )$.In this case the gravitational field with constant acceleration $\vec{g}=-g\vec{e_{3}}$ acts on the fluids, here $g>0$ and $\vec{e_{3}}$ is unit vector of the vertical axis $Ox_{3}$, which is directed opposite to $\vec{g}$.
Let us consider the basic case of stable stratification of the fluids on densities $\rho _{0k}=\rho _{0k}\left ( x_{3} \right )$ $\left ( k=1,2 \right )$ :
$0< N_{k, min}^{2}\leq N_{k}^{2}\left ( x_{3} \right )\leq N_{k, max}^{2}=:N_{0, k}^{2}< \infty $,
$N_{k}^{2}\left ( x_{3} \right ):=-\frac{g{\rho}'_{0k}\left ( x_{3} \right )}{\rho _{0k}\left ( x_{3} \right )}$, $\left ( k=1,2 \right )$.
The problem on small motions of the system of viscous stratification fluids is investigated on base of an approach connected with application of so-called operator matrices theory with unbounded entries and general theory of the abstract operator-differential equations. Existence conditions of strong solution of initial boundary value are obtained.

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