Solvability Of Initial-boundary Value Problem Research Articles

Navier–Stokes Equation in a Cone with Cross-Sections in the Form of 3D Spheres, Depending on Time, and the Corresponding Basis

The work establishes the unique solvability of a boundary value problem for a 3D linearized system of Navier–Stokes equations in a degenerate domain represented by a cone. The domain degenerates at the vertex of the cone at the initial moment of time, and, as a consequence of this fact, there are no initial conditions in the problem under consideration. First, the unique solvability of the initial-boundary value problem for the 3D linearized Navier–Stokes equations system in a truncated cone is established. Then, the original problem for the cone is approximated by a countable family of initial-boundary value problems in domains represented by truncated cones, which are constructed in a specially chosen manner. In the limit, the truncated cones will tend toward the original cone. The Faedo–Galerkin method is used to prove the unique solvability of initial-boundary value problems in each of the truncated cones. By carrying out the passage to the limit, we obtain the main result regarding the solvability of the boundary value problem in a cone.

Solvability of initial-boundary value problem for the modified Kelvin–Voigt model with memory along trajectories of fluid motion

Solvability of initial-boundary value problem for the modified Kelvin–Voigt model with memory along trajectories of fluid motion

Differential-difference systems in the analysis of weak solvability of initial-boundary value problems with a spatial variable in a network-like domain

In the work, the approach and the corresponding methods, which make it possible to construct a priori estimates of weak solutions of a differential-difference system with a spatial variable varying in a multidimensional network-like domain are indicated. Such estimates in spaces of summable functions are used to find solvability conditions for boundary value problems of various types for differential-difference systems. In addition, a priori estimates are used to justify the application of the method of discretization with respect to the time variable (semi-discretization) to the analysis of the weak solvability of initial-boundary value problems and the subsequent construction of approximations of weak solutions. The rationale for the approach used is the fact that in a fairly wide class applied analysis of the problems of transporting continuous media networks-like carriers, the representation of mathematical models of the process using the formalisms of differential-difference systems is the only tool for effectively solving these problems. For example, the reduction of a differential system (initial-boundary value problem) to the corresponding differential-difference system makes it possible not only to significantly simplify the analysis of problems of optimal control of a differential system (since this analysis reduces to studying the problem of optimal control of a system of elliptic equations), but also, using classical methods of control theory for elliptic systems, algorithmize the original problem. The reduction used often facilitates establishing the conditions for the existence and uniqueness of optimal control of a differential system. These problems also include a fairly large range of studies of non-stationary network-like hydrodynamic processes and flow phenomena. As an illustration of the approach used and the results obtained, the analysis of the solvability of the linearized Navier–Stokes system is given and the ways of studying the nonlinear Navier–Stokes system are indicated.

Optimal control of thermal and wave processes in composite materials

The paper indicates the approach and the corresponding to it method of penalty functions for analyzing the problems of optimal control of thermal and wave processes in structural elements made of composite materials (composites). An object that is quite common in the industrial sphere, the structure of which is a set of layers (phases) of unidirectional composites — layered composites, is considered. When solving problems related to the analysis and description of the states of composites, quantitative characteristics of layers that are not functions of the coordinates of the points of the medium are usually used in order not to solve the corresponding problems for an inhomogeneous medium. Such functions are elements of Sobolev spaces, first of all, functions summable with a square. The convenience lies in the fact that when finding the conditions for solvability of initial-boundary value problems of various types (in most cases, such problems are the basis of mathematical models of many physical processes), it is possible to reduce to operator-difference systems, for which it is easy to construct a priori estimates of weak solutions. The next step after establishing the weak solvability of the initial-boundary value problem of the thermal or wave process in composites is the formulation and solution of the problem of optimal control of these processes. The proposed method of penalty functions on the example of solving such problems is a general method. It is applicable with slight modifications also not only in the case of elliptic, parabolic and other problems (including nonlinear) for scalar functions, but also for vector functions. An example of the latter is the Navier–Stokes system, widely used in the description of network-like hydrodynamic processes, considered in Sobolev spaces, the elements of which are functions with carriers on n-dimensional network-like domains, n greater or equal to 2.

On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions

The problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated and studied. It is proved a theorem of existence and uniqueness of a weak solution for the problem under consideration.

Solvability Issues of a Pseudo-Parabolic Fractional Order Equation with a Nonlinear Boundary Condition

This paper is devoted to the fundamental problem of investigating the solvability of initial-boundary value problems for a quasi-linear pseudo-parabolic equation of fractional order with a sufficiently smooth boundary. The difference between the studied problems is that the boundary conditions are set in the form of a nonlinear boundary condition with a fractional differentiation operator. The main result of this work is establishing the local or global solvability of stated problems, depending on the parameters of the equation. The Galerkin method is used to prove the existence of a quasi-linear pseudo-parabolic equation’s weak solution in a bounded domain. Using Sobolev embedding theorems, a priori estimates of the solution are obtained. A priori estimates and the Rellich–Kondrashov theorem are used to prove the existence of the desired solutions to the considered boundary value problems. The uniqueness of the weak generalized solutions of the initial boundary value problems is proved on the basis of the obtained a priori estimates and the application of the generalized Gronwall lemma. The need to consider and study such initial boundary value problems for a quasi-linear pseudo-parabolic equation follows from practical requirements, such as solving fractional differential equations that simulate physical processes that occur during the study of liquid filtration processes, etc.

Blowing-up solutions of the time-fractional dispersive equations

Abstract This paper is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

Equivalence of Weak Solvability of Initial-Boundary Value Problems for the Jeffries-Oldroyd Model and one Integro-Differential System with Memory

The equivalence of weak solvability of initial boundary value problems for the Jeffries-Oldroyd model and one integro-differential system with memory is established. The proofs of the statements are essentially based on the properties of regular Lagrangian flows.

Эквивалентность слабой разрешимости начально-краевых задач для модели Джеффриса--Олдройда и одной интегродифференциальной системы с памятью

Эквивалентность слабой разрешимости начально-краевых задач для модели Джеффриса--Олдройда и одной интегродифференциальной системы с памятью

РОЗВ’ЯЗУВАНIСТЬ ОСЕРЕДНЕНИХ ЗАДАЧ У ЗГОРТКАХ ДЛЯ СЛАБО ПОРИСТИХ СЕРЕДОВИЩ

Initial boundary value problems for nonstationary equations of diffusion and filtration in weakly porous media are considered. Assertions about the solvability of such problems and the corresponding homogenized problems with convolutions are given. These statements are proved for general initial data and inhomogeneous initial conditions and are generalizations of classical results on the solvability of initial-boundary value problems for the heat equation. The proofs use the methods of a priori estimates and the well-known Agranovich–Vishik method, developed to study parabolic problems of general type.

Solvability of initial-boundary value problem of a multiple characteristic fifth-order operator-differential equation

In this study, we establish existence-uniqueness of a vector function in appropriate Sobolev-type space for a boundary value problem of a fifth-order operator differential equation. Proper conditions are obtained for the given problem to be well-posed. Much effort is devoted to develop the association between these conditions and the operator coefficients of the investigated equation. In this paper, accurate estimates of the norms of the intermediate derivatives operators are presented and used to determine the solvability conditions.

Local Solvability of Initial-Boundary Value Problem for One-Dimensional Equations of Polytropic Flows of Viscous Compressible Multifluids

We consider the initial-boundary value problem governing unsteady polytropic motions of viscous compressible multifluids. We prove the existence and uniqueness of a strong solution to the problem. Bibliography: 11 titles.

Analytic in a Sector Resolving Families of Operators for Degenerate Evolution Fractional Equations

We introduce a class of pairs of operators defining a linear homogeneous degenerate evolution fractional differential equation in a Banach space. Reflexive Banach spaces are represented as the direct sums of the phase space of the equation and the kernel of the operator at the fractional derivative. In a sector of the complex plane containing the positive half-axis, we construct an analytic family of resolving operators that degenerate only on the kernel. The results are used in the study of the solvability of initial-boundary value problems for partial differential equations containing fractional time-derivatives and polynomials in the Laplace operator with respect to the spatial variable.

Correctness of the initial-boundary problem of the compressible fluid filtration in a viscous porous medium

The local solvability of initial-boundary value problem for the system of the equations of non stationary fluid motion in a viscous deformable medium in the field of gravity is proved.

On the solvability of initial-boundary value problems for a viscous compressible fluid in an infinite time interval

On the solvability of initial-boundary value problems for a viscous compressible fluid in an infinite time interval

Global Solvability of Initial Boundary-Value Problems for Nonlinear Analogs of the Boussinesq Equation

В работе исследуется разрешимость естественных (первой, второй и смешанной) начально-краевых задач для нелинейных аналогов уравнения Буссинеска. Доказываются теоремы единственности регулярных решений, а также теоремы о разрешимости в целом. Библиография: 20 названий.

Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation

The solvability of the natural (first, second, and mixed) initial boundary-value problems for nonlinear analogs of the Boussinesq equation is studied. Uniqueness theorems for regular solutions and global solvability theorems are proved.

Well-posed solvability of functional-differential equations with unbounded operator coefficients

We study functional-differential equations with unbounded variable operator coefficients and variable delays in a Hilbert space. We prove the well-posed solvability of initial-boundary value problems for the above-mentioned equations in Sobolev spaces of vector functions on the positive half-line.

On the solvability of initial boundary-value problems for a class of operator-differential equations of third order

We obtain sufficient conditions for the regular solvability of initial boundary-value problems for a class of operator-differential equations of third order with variable coefficients on the semiaxis. These conditions are expressed only in terms of the operator coefficients of the equations under study. We obtain estimates of the norms of intermediate derivative operators via the discontinuous principal parts of the equations and also find relations between these estimates and the conditions for regular solvability.

Dynamics of a rotating layer of an ideal electrically conducting incompressible inhomogeneous fluid in an equatorial region

The equations describing the three-dimensional equatorial dynamics of an ideal electrically conducting inhomogeneous rotating fluid are studied. The magnetic and velocity fields are represented as superpositions of unperturbed steady-state fields and those induced by wave motion. As a result, after introducing two auxiliary functions, the equations are reduced to a special scalar one. Based on the study of this equation, the solvability of initial-boundary value problems arising in the theory of waves propagating in a spherical layer of an electrically conducting density-inhomogeneous rotating fluid in an equatorial zone is analyzed. Particular solutions of the scalar equation are constructed that describe small-amplitude wave propagation.