Vol 69, No 4 (2023)

Articles

On ellipticity of operators with shear mappings

Boltachev A.V.

Abstract

The nonlocal boundary value problems are considered, in which the main operator and the operators in the boundary conditions include the differential operators and twisting operators. The de nition of the trajectory symbols for this class of problems is given. We show that the elliptic problems de ne the Fredholm operators in the corresponding Sobolev spaces. The ellipticity condition of such nonlocal boundary value problem is given.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):565-577
pages 565-577 views

Stationary states in population dynamics with migration and distributed offspring

Davydov A.A., Khachatryan K.A.

Abstract

For an integral equation whose solutions provide stationary states of a population distributed in an arithmetic space, we nd the conditions for the existence of its solution and conditions under which this equation has no more than one solution.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):578-587
pages 578-587 views

Exponential stability of the flow for a generalized Burgers equation on a circle

Djurdjevac A., Shirikyan A.R.

Abstract

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the \(L^1\) norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on \(R\), which is exponentially stable in \(H^1\) as \(t\to+\infty\). In the case of a random external force, we show that the difference between two trajectories goes to zero with probability \(1\).

Contemporary Mathematics. Fundamental Directions. 2023;69(4):588-598
pages 588-598 views

Eta-invariant of elliptic parameter-dependent boundary-value problems

Zhuikov K.N., Savin A.Y.

Abstract

In this paper, we study the eta-invariant of elliptic parameter-dependent boundary value problems and its main properties. Using Melrose’s approach, we de ne the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundary value problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary value problems and is based on the reduction of parameter-dependent boundary value problems to boundary value problems with no parameter.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):599-620
pages 599-620 views

The existence problem of feedback control for one fractional Voigt model

Zvyagin A.V., Kostenko E.I.

Abstract

In this paper, we study the feedback control problem for a mathematical model that describes the motion of a viscoelastic fluid with memory along velocity eld trajectories. We prove the existence of an optimal control that gives a minimum to a given bounded and semi-continuous from below quality functional. The proof uses the approximation-topological approach, the theory of regular Lagrangian flows, and the theory of topological degree for multivalued vector elds.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):621-642
pages 621-642 views

Einstein material balance and modeling of the flow of compressible fluid near the boundary

Ibraguimov A., Zakirov E., Indrupskiy I., Anikeev D., Zhaglova A.

Abstract

We consider sewing machinery between nite difference and analytical solutions de ned at different scales: far away and near the source of the perturbation of the flow. One of the essences of the approach is that the coarse problem and the boundary-value problem in the proxy of the source model two different flows. In his remarkable paper, Peaceman proposes a framework for dealing with solutions de ned on different scales for linear time independent problems by introducing the famous Peaceman well block radius. In this article, we consider a novel problem: how to solve this issue for transient flow generated by the compressibility of the fluid. We are proposing a method to glue solution via total fluxes, which are prede ned on coarse grid, and changes in pressure, due to compressibility, in the block containing production (injection) well. It is important to mention that the coarse solution “does not see” the boundary. From an industrial point of view, our report provides a mathematical tool for the analytical interpretation of simulated data for compressible fluid flow around a well in a porous medium. It can be considered a mathematical “shirt” on the Peaceman well-block radius formula for linear (Darcy) transient flow but can be applied in much more general scenarios. In the article, we use the Einstein approach to derive the material balance equation, a key instrument to de ne R0. We will expand the Einstein approach for three regimes of Darcy and non-Darcy flows for compressible fluids (time-dependent): 1. stationary; 2. pseudostationary; 3. boundary dominated. Note that in all known authors literature, the rate of production on the well is time-independent.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):643-663
pages 643-663 views

Boundary-value problems for differential-difference equations with nite and in nite orbits of boundaries

Ivanova E.P.

Abstract

We consider boundary-value problems for differential-difference equations containing incommensurable shifts of arguments in the higher-order terms. We show that for the case when the orbits of the domain boundary generated by the set of shifts of the difference operator are nite, the original problem is similar to the boundary-value problem for differential-difference equations with integer shifts of arguments. The case of an in nite boundary orbit is also studied.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):664-675
pages 664-675 views

On the structure of weak solutions of the Riemann problem for a degenerate nonlinear diffusion equation

Panov E.Y.

Abstract

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coe cient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a nite number of variables. A similar result is true for Stefan’s problem. In the limit, when the number of phases tends to in nity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):676-684
pages 676-684 views

On plane oscillations of the cold plasma in a constant magnetic field

Rozanova O.S.

Abstract

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Contemporary Mathematics. Fundamental Directions. 2023;69(4):685-696
pages 685-696 views

Boundary-value problem for an elliptic functional differential equation with dilation and rotation of arguments

Rossovskii L.E., Tovsultanov A.A.

Abstract

The paper is devoted to the Dirichlet problem in a flat bounded domain for a linear secondorder functional differential equation in the divergent form with dilation, contraction and rotation of the argument of the higher-order derivatives of the unknown function. We study the existence, the uniqueness and the smoothness of the generalized solution for all possible values of the coefficients and parameters of transformations in the equation.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):697-711
pages 697-711 views

On the existence of time-periodic solutions of nonlinear parabolic differential equations with nonlocal boundary conditions of the Bitsadze-Samarskii type

Solonukha O.V.

Abstract

We study a nonlinear parabolic differential equation in a bounded multidimensional domain with nonlocal boundary conditions of the Bitsadze-Samarskii type. We prove existence theorems for a periodic in time generalized solution. Su cient conditions for the existence of generalized solutions contain either an algebraic ellipticity condition or an algebraic strong ellipticity condition for the auxiliary differential-difference operator.

Contemporary Mathematics. Fundamental Directions. 2023;69(4):712-725
pages 712-725 views

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