# Vol 64, No 3 (2018): Proceedings of the Crimean Autumn Mathematical School-Symposium

**Year:**2018**Articles:**5**URL:**https://journals.rudn.ru/CMFD/issue/view/1254**DOI:**https://doi.org/10.22363/2413-3639-2018-64-3

## Full Issue

## New Results

### Inverse Spectral Problem for Integrodiﬀerential Sturm-Liouville Operators with Discontinuity Conditions

#### Abstract

We consider the Sturm-Liouville operator perturbed by a convolution integral operator on a ﬁnite interval with Dirichlet boundary-value conditions and discontinuity conditions in the middle of the interval. We study the inverse problem of restoration of the convolution term by the spectrum. The problem is reduced to solution of the so-called main nonlinear integral equation with a singularity. To derive and investigate this equations, we do detailed analysis of kernels of transformation operators for the integrodiﬀerential expression under consideration. We prove the global solvability of the main equation, this implies the uniqueness of solution of the inverse problem and leads to necessary and suﬃcient conditions for its solvability in terms of spectrum asymptotics. The proof is constructive and gives the algorithm of solution of the inverse problem.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(3):427-458

427-458

### Operator Approach to the Problem on Small Motions of an Ideal Relaxing Fluid

#### Abstract

In this paper, we study the problem on small motions of an ideal relaxing ﬂuid that ﬁlls a uniformly rotating or ﬁxed container. We prove a theorem on uniform strong solvability of the corresponding initial-boundary value problem. In the case where the system does not rotate, we ﬁnd an asymptotic behavior of the solution under the stress of special form. We investigate the spectral problem associated with the system under consideration. We obtain results on localization of the spectrum, on essential and discrete spectrum, and on spectral asymptotics. For nonrotating system in zero-gravity conditions we prove the multiple basis property of a special system of elements. In this case, we ﬁnd an expansion of the solution of the evolution problem in the special system of elements.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(3):459-489

459-489

490-546

### To the Problem on Small Motions of the System of Two Viscoelastic Fluids in a Fixed Vessel

#### Abstract

In this paper, we study the problem of small motions of two Oldroyd viscoelastic incompressible ﬂuids contained in a ﬁxed vessel. By means of the operator approach, we reduce the original initialboundary value problem to the Cauchy problem for a diﬀerential operator equation in a Hilbert space and prove the well-posed solvability of the problem on an arbitrary interval of time. We obtain the equation for normal oscillations of the hydraulic system under consideration (Krein generalized operator pencil).

**Contemporary Mathematics. Fundamental Directions**. 2018;64(3):547-572

547-572

### Small Motions of an Ideal Stratiﬁed Fluid in a Basin Covered with Ice

#### Abstract

We study the problem on small motions of an ideal stratiﬁed ﬂuid with a free surface partially covered with crushed ice. The crushed ice is supposed to be ponderable particles of some matter ﬂoating on the free surface. These particles do not interact with each other during oscillations of the free boundary (or this interaction is neglible) and stay on the surface during these oscillations. Using the method of orthogonal projecting of boundary-value conditions on the free surface and introducing auxiliary problems, we reduce the original initial-boundary value problem to the equivalent Cauchy problem for a second-order diﬀerential equation in some Hilbert space. We obtain conditions under which there exists a strong with respect to time solution of the initial-boundary value problem describing the evolution of this hydraulic system.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(3):573-590

573-590