Volume 233, Nº 6 (2018)
- Ano: 2018
- Artigos: 9
- URL: https://journals.rcsi.science/1072-3374/issue/view/14951
Article
Nonstationary Problem of Complex Heat Transfer in a System of Semitransparent Bodies with Boundary-Value Conditions of Diffuse Reflection and Refraction of Radiation
Resumo
We consider a nonstationary initial boundary-value problem describing complex (radiative-conductive) heat transfer in a system of semitransparent bodies. To describe radiation propagation, we use the radiation transfer equation with boundary-value conditions of diffuse reflection and refraction of radiation. 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 as well as its regularity are obtained.
On the Stabilization Rate of Solutions of the Cauchy Problem for a Parabolic Equation with Lower-Order Terms
Resumo
The following Cauchy problem for parabolic equations is considered in the half-space \( \overline{D}={\mathrm{\mathbb{R}}}^N\times \left[0,\infty \right) \), N ≥ 3:
It is proved that for any bounded and continuous in ℝN initial function u0(x), the solution of the above Cauchy problem stabilizes to zero uniformly with respect to x from any compact set K in ℝN either exponentially or as a power (depending on the estimate for the coefficient c(x, t) of the equation).
Continuous Dependence on Translations of the Independent Variable for Solutions of Boundary-Value Problems for Differential-Difference Equations
Resumo
We consider boundary-value problems for differential-difference operators with perturbations in translations of the independent variable. We prove that the family of differential-difference operators is positive definite uniformly with respect to translations of the independent variable. Solutions of such problems depend continuously on these translations. We consider the coercivity problem for differential-difference operators with incommensurable translations of the independent variable and study the approximation of such operators by rational operators.
On Stability of Perturbed Semigroups in Partially Ordered Banach Spaces
Resumo
We prove necessary and sufficient stability conditions for perturbed semigroups of linear operators in Banach spaces with cones and consider examples using these conditions. In particular, we consider an example where the boundary-value problem is perturbed by a linear operator with a delayed independent variable and establish stability conditions for such a perturbed semigroup.
Differential Equations with Degenerate Operators at the Derivative Depending on an Unknown Function
Resumo
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 an irreversible operator. For simplicity, in Secs. 1–3, the geometric multiplicity of λ0 is equal to one, i.e., dimN(A0) = 1, N(A0) = span{φ}, dimN*(\( {A}_0^{\ast } \)) = 1, N*(\( {A}_0^{\ast } \)) = span{ψ}, and it is assumed that the operator function A(λ) is linear with respect to λ. In Sec. 4, the polynomial dependence of A(λ) is linearized. However, the results of existence theorems for bifurcations are obtained for the case where there are several Jordan chains. Applications to degenerate differential equations of the form [A0 + R(·, x)]x′= Bx are provided.
Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview
Resumo
In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws
where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form
where αj and \( {\alpha}_j^{\prime } \) are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form).
The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified 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.
Elliptic G-Operators on Manifolds with Isolated Singularities
Resumo
In the present work we study elliptic operators on manifolds with singularities in the situation where the manifold is endowed with an action of a discrete group G. As usual in elliptic theory, the Fredholm property of an operator is governed by the properties of its principal symbol. We show that the principal symbol in our situation is a pair consisting of the symbol on the main stratum (interior symbol) and the symbol at the conical point (conormal symbol). The Fredholm property of elliptic elements is obtained.
Magnetic Schrödinger Operator from the Point of View of Noncommutative Geometry
Resumo
We give an interpretation of the magnetic Schrödinger 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 the magnetic Schrödinger operator. We show how this idea can be applied to the integer quantum Hall effect.
Stability of Solutions of Initial Boundary-Balue Problems of Aerohydroelasticity
Resumo
In designing structures and devices interacting with the flow 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 definition of stability of an elastic body taken in the article corresponds to Lyapunov’s concept of stability of a dynamical system. On the base of the proposed nonlinear mathematical model the dynamic stability of the elastic aileron of the wing taking into account the incident subsonic flow of gas or liquid (in an ideal model of an incompressible environment) is investigated. Also a nonlinear mathematical model of the device relating to the vibration technique, which is intended for intensification of technological processes, for example, the process of mixing, is considered. The action of these devices is based on the oscillations of elastic elements during the gas or liquid flow. The dynamic stability of the elastic element located on one wall of the flow channel with the subsonic flow of gas or liquid (in an ideal model of a compressible environment) is investigated. Both models are described by a coupled nonlinear system of differential 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, we obtain the sufficient conditions of the stability, the imposed restrictions on the free-stream velocity of the gas, the flexural stiffness of the elastic element, and other parameters of the mechanical system. The examples of construction of the stability regions for particular parameters of the mechanical system are presented.