# On Small Movements of a System “fluid–gas” in a Bounded Region

In the paper, we consider a problem on small motions of a system of viscoelastic ﬂuid and gas in a stationary container. One of models of such viscoelastic ﬂuid is Oldroid’s model. It is described, for example, in the book Eirich, F. R. Rheology. Theory and Applications. New York: Academic Press, 1956. It should be noted that the present paper is based on the previous N. D. Kopachevsky works together with Azizov, T. Ya., Orlova L. D., Krein, S. G. Namely, problem on small movements of one viscoelastic ﬂuid for generalized Oldroid’s model, small motions of a viscoelastic ﬂuid in an open container, oscillations of a system of ideal ﬂuids were investigated in these papers.

The aim of this paper is to use an operator approach of mentioned works, to develop new approach and to prove the theorem on strong solvability for initial-boundary-value problem generated by a problem on small motions of a system of viscoelastic ﬂuid and gas in ani mmovable container.

This paper is organized as follows. Section 1 is an introduction. In section 2 we formulate mathematical statement of the problem: linearized equations of movements, stickiness condition, kinematic and dynamic conditions. Further, in this section we receive the law of full energy balance. In section 3 we choose the functional spaces generated by the problem for each ﬂuid. For applying of method of orthogonal projection we need to choose orthogonal decomposition on corresponding spaces. Section 4 is devoted to the method of orthogonal projection which allow us to get new statement of the problem without some trivial equations. Important part of section 4 is formulation of auxiliary problems which help us to make transition to the Cauchy problem for a system of integro-differential equation in some Hilbert space. In section 5 we reduce this problem to a system of integro-differential equation. This system can be rewrite in section 6 as operator integro-differential equation in the sum of Hilbert spaces. Properties of main operator of this problem are studied in section 6 too. Transition to operator differential equation in the sum of Hilbert spaces is realized in section 7. Section 8 is devoted to the existence and uniqueness theorems for ﬁnal operator differential equation as for original initial boundary-value problem. This result based on factorization, closure and accretivity property of operator matrix. Finally, in section 9 we consider the spectral problem on normal oscillations corresponding to the evolution problem.

**Keywords:** viscoelastic ﬂuid, gas, hydrodynamic system, orthogonal projector, Cauchy problem.