# Vol 59, No (2016)

**Year:**2016**Articles:**9**URL:**https://journals.rudn.ru/CMFD/issue/view/1590

## Full Issue

## Articles

### Nonstationary Problem of Complex Heat Transfer in a System of Semitransparent Bodies with Radiation Di use Re ection and Refraction Boundary-Value Conditions

#### Abstract

We consider a nonstationary initial-boundary value problem describing complex (radiative-con-ductive) heat transfer in a system of semitransparent bodies. To describe radiation propagation, we use the transport equation with radiation di use re ection and refraction boundary-value conditions. We take into account that the radiation intensity and optical properties of bodies depend on the radiation frequency. The unique solvability of a weak solution is established. The comparison theorem is proved. A priori estimates of a weak solution are obtained as well as its regularity.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:5-34

5-34

### Stability of Solutions of Initial Boundary Value Problems of Aerohydroelasticity

#### Abstract

At designing structures and devices interacting with the ow of gas or liquid, it is necessary to solve the problems associated with the investigation of the stability required for their functioning and operational reliability. The de nition of stability of an elastic body, taken in the article, corresponds to the Lyapunov’s concept of stability of dynamical system. On the base of a proposed nonlinear mathematical model the dynamic stability of the elastic aileron of the wing taking into account the incident subsonic ow of gas or liquid (in an ideal model of a incompressible environment) is investigated. Also a nonlinear mathematical model of the device relating to the vibration technique, which is intended for intensi cation of technological processes, for example, the process of mixing is considered. The action of these devices is based on the oscillations of elastic elements at the owing around a of gas or liquid ow. The dynamic stability of the elastic element, located on one wall of the ow channel with the subsonic ow of gas or liquid (in an ideal model of a compressible environment) is investigated. The both models is described by coupled nonlinear system of di erential equations for the unknown functions - the potential of the gas velocity and deformation of the elastic element. On the basis of the construction of functionals, the su cient conditions of the stability, impose restrictions on the free-stream velocity of the gas, the exural sti ness of the elastic element, and other parameters of the mechanical system is obtained. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:35-52

35-52

### On the Stabilization Rate of Solutions of the Cauchy Problem for a Parabolic Equation with Lower-Order Terms

#### Abstract

For a parabolic equation in the half-space D = RN × [0, ∞), N >= 3, we consider the Cauchy problem L1u ≡ Lu + c(x, t)u - ut = 0, (x, t) ∈ D, u(x, 0) = u0(x), x ∈ RN . Depending on estimates on the coe cient c(x, t), we establish power or exponential rate of stabilization of solutions of the Cauchy problem равномерно по x на каждом компакте K в RN для произвольной ограниченной непрерывной в RN начальной функции u0(x).

**Contemporary Mathematics. Fundamental Directions**. 2016;59:53-73

53-73

### Continuous Dependence of Solutions of Boundary-Value Problems for Di erential-Di erence Equations on Shifts of the Argument

#### Abstract

We consider boundary-value problems for di erential-di erence operators with perturbations in shifts of the argument. We prove that the family of di erential-di erence operators is positive de nite uniformly with respect to the shifts of the argument. Solutions of such problems depend continuously on these shifts. We consider the coercivity problem for di erential-di erence operators с with incommensurable shifts of the argument and study the approximation of such operators by rational operators.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:74-96

74-96

### On Stability of Perturbed Semigroups in Partially Ordered Banach Spaces

#### Abstract

We prove necessary and su cient conditions for stability of perturbed semigroups of linear operators in Banach spaces with cones and consider some examples of using these conditions. In particular, we consider an example where the boundary-value problem is perturbed by a linear operator with delayed argument and establish conditions of stability for such a perturbed semigroup.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:97-118

97-118

### Di erential Equations with Degenerate, Depending on the Unknown Function Operator at the Derivative

#### Abstract

We develop the theory of generalized Jordan chains of multiparameter operator functions A(λ) : E1 → E2, λ ∈ Λ, dimΛ = k, dimE1 = dimE2 = n, where A0 = A(0) is a noninvertible operator. To simplify the notation, in Secs. 1-3 the geometric multiplicity λ0 is set to 1, i. e. dimN(A0) = 1, N(A0) = span{ϕ}, dimN∗(A∗0) = 1, N∗(A∗0) = span{ψ}, and the operator function A(λ) is supposed to be linear with respect to λ. For the polynomial dependence of A(λ), in Sec. 4 we consider a linearization. However, the bifurcation existence theorems hold in the case of several Jordan chains as well. We consider applications to degenerate differential equations of the form [A0 + R(·, x)]x*= Bx.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:119-147

119-147

### Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview

#### Abstract

The aim of this paper is to provide the reader with a proof of such quadratic estimate in a simpli ed setting, in which: - all the main ideas of the construction are presented; - all the technicalities of the proof in the general setting [8] are avoided.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:148-172

148-172

### Elliptic G-Operators on Manifolds with Isolated Singularities

#### Abstract

We study elliptic operators on manifolds with singularities such that a discrete group G acts on the manifold. Following the standard elliptic theory approach, we de ne the Fredholm property of an operator by its principal symbol. For this problem, we prove that the symbol is a pair consisting of the symbol on the principal stratum (the inner symbol) and the symbol at the conical point (the conormal symbol). We establish the Fredholm property of elliptic elements.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:173-191

173-191

### Magnetic Schro¨dinger Operator from the Point of View of Noncommutative Geometry

#### Abstract

We give an interpretation of magnetic Schro¨dinger operator in terms of noncommutative geometry. In particular, spectral properties of this operator are reformulated in terms of C∗-algebras. Using this reformulation, one can employ the machinery of noncommutative geometry, such as Hochschild cohomology, to study the properties of magnetic Schro¨dinger operator. We show how this idea can be applied to the integer quantum Hall e ect.

**Contemporary Mathematics. Fundamental Directions**. 2016;59:192-200

192-200