Initial-boundary Value Problem For System Research Articles

New Thought on Matsumura–Nishida Theory in the L_p–L_q Maximal Regularity Framework

This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier–Stokes equations describing the motion of viscous, compressible, barotropic fluid flows in a three dimensional exterior domain with non-slip boundary conditions. This was first proved by an excellent paper due to Matsumura and Nishida (Commun Math Phys 89:445–464, 1983). In [10], they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to L_2 in space-time, detailed statement of Matsumura and Nishida theorem is given in Theorem 1 of Sect. 1 of context. This requirement is essentially used to estimate the L_infty norm of necessary order of derivatives in order to enclose the iteration scheme with the help of Sobolev inequalities and also to treat the material derivatives of the mass density. On the other hand, this paper gives the global wellposedness of the same problem as in [10] in L_p (1 <p le 2) in time and L_2cap L_6 in space maximal regularity class, which is an improvement of the Matsumura and Nishida theory in [10] from the point of view of the minimal requirement of the regularity of solutions. In fact, after changing the material derivatives to time derivatives by Lagrange transformation, enough estimates obtained by combination of the maximal L_p (1 <p le 2) in time and L_2cap L_6 in space regularity and L_p–L_q decay estimate of the Stokes equations with non-slip conditions in the compressible viscous fluid flow case enable us to use the standard Banach’s fixed point argument. Moreover, one of the purposes of this paper is to present a framework to prove the L_p–L_q maximal regularity for parabolic-hyperbolic type equations with non-homogeneous boundary conditions and how to combine the maximal L_p–L_q regularity and L_p–L_q decay estimates of linearized equations to prove the global well-posedness of quasilinear problems in unbounded domains, which gives a new thought of proving the global well-posedness of initial-boundary value problems for systems of parabolic or parabolic-hyperbolic equations appearing in mathematical physics.

Investigation of dynamic processes in pressure measurement systems for gas-liquid media

Initial-boundary value problems for systems of differential equations are considered, which are mathematical models of the mechanical system "pipeline - pressure sensor". In such a system, to mitigate the effects of vibration accelerations and high temperatures, the sensor is located at a certain distance from the engine and is connected to it via a pipeline. The "pipeline - pressure sensor" system is designed to measure pressure in gas-liquid media, for example, to control the pressure of the working medium in the combustion chambers of engines. On the basis of the proposed models, the joint dynamics of the sensitive element of the pressure sensor and the working medium in the pipeline is studied. To describe the motion of the working medium, linear models of fluid and gas mechanics are used, to describe the dynamics of a sensitive element, linear models of the mechanics of a deformable solid are applied. Analytical and numerical methods for solving initial-boundary value problems under study are presented. The numerical study of the initial-boundary value problem was carried out on the basis of the Galerkin method. In analytical study using the introduction of averaged characteristics, the solution of the original two-dimensional problem is reduced to the study of a one-dimensional model, whose further study made it possible to reduce the solution of the problem to the study of a differential equation with a deviating argument. Also, a numerical experiment is carried out and an example of calculating the deflection of the sensor’s moving element is presented.

Mathematical modeling in problems about dynamics and stability of elastic elements of wing profiles

The mathematical models describing the dynamics of elastic elements of wing structures and representing the initial-boundary value problems for systems of partial differential equations are proposed. The dynamics and stability of elastic elements of wings, flown around by a gas or liquid stream in a model of an incompressible medium, are investigated. To study the dynamics of elastic elements and a gas-liquid medium, both linear and nonlinear models of the mechanics of a solid deformable body and linear models of the mechanics of liquid and gas are used. On the basis of the constructed functionals for partial differential equations, the sufficient stability conditions are obtained in analytical form. The conditions impose restrictions on the parameters of mechanical systems. The obtained stability conditions are necessary for solving the problems of controlling the parameters of the aeroelastic system. On the basis of the Galerkin method, a numerical study of the dynamics of elastic elements was carried out, the reliability of which is confirmed by the obtained analytical results.

Mathematical modeling of pressure measurement systems in gas-liquid media

The article discusses the initial-boundary value problems for systems of differential equations, which are mathematical models of the mechanical system "pipeline - pressure sensor that is designed to measure pressure in gas-liquid media. On the basis of the proposed models, the joint dynamics of the pressure sensor sensitive element and of the working medium in the pipeline connecting the sensor to the combustion chamber of the engine is investigated. To describe the movement of the working medium, linear models of the mechanics of liquid and gas are used; to describe the dynamics of the sensitive element, both linear and nonlinear models of the mechanics of a solid deformable body are used. The solutions of stated initial-boundary value problems are carried out on the basis of the Galerkin method and the finite-difference method.

An Approach to the Solution of the Initial Boundary-Value Problem for Systems of Fourth-Order Hyperbolic Equations

The initial boundary-value problem for systems of fourth-order partial differential equations with two independent variables is considered. By using a new unknown eigenfunction, the problem under consideration is reduced to an equivalent nonlocal problem for a system of second-order hyperbolic-type integro-differential equations with integral conditions. An algorithm for finding an approximate solution of the resulting equivalent problem is proposed, and its convergence is proved. Conditions for the existence of a unique classical solution of the initial boundary-value problem for systems of fourth-order differential equations are established in terms of the coefficients of the system and the boundary matrices.

The Existence Theorem for a Weak Solution to Initial-Boundary Value Problem for System of Equations Describing the Motion of Weak Aqueous Polymer Solutions

In this paper, we prove the existence of weak solutions to the initial-boundary value problem that describes the motion of weakly concentrated aqueous solutions of polymers. The proof is based on the approximation-topological approach. At the first step, we approximate the operator equation, which is equivalent to the weak statement of the considered problem, with another operator equation with “good” properties and prove the solvability of the latter equation. At the second step, we proceed to the limit, i.e., we prove that the sequence of solutions contains a subsequence that converges weakly to the solution of the initial problem as the approximation parameter tends to zero.

Теорема существования слабого решения начально-краевой задачи для системы уравнений, описывающей движение слабых водных растворов полимеров

Теорема существования слабого решения начально-краевой задачи для системы уравнений, описывающей движение слабых водных растворов полимеров

An efficient asymptotic-numerical method to solve nonlinear systems of one-dimensional balance laws

An asymptotic-numerical method to solve the initial-boundary value problems for systems of balance laws in one space dimension, on the half space is developed. Expansions in powers of $$t^{-1/2}$$ are used, in view of the precise asymptotic behavior recently established on theoretical bases. This approach increases considerably the efficiency of a previous one, where just expansions in inverse powers of t were made. Numerical examples and comparisons with the Godunov, the asymptotic high order, and the asymptotic-numerical method earlier developed are presented. Expanding the solution in powers of $$t^{-1/2}$$ instead of $$t^{-1}$$ , a saving of about one-half of the CPU time can be realized, still achieving the same accuracy.

Phase space of the initial-boundary value problem for the Oskolkov system of nonzero order

The phase space of the Dirichlet initial-boundary value problem for a system of partial differential equations modeling the flow of an incompressible viscoelastic Kelvin-Voigt fluid of nonzero order is described. The investigation is based on the theory of semilinear Sobolev-type equations and the concepts of a relatively spectral bounded operator and a quasi-stationary trajectory for the corresponding Oskolkov system modeling the plane-parallel flow of the above fluid.

Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws

Solutions of initial-boundary value problems for systems of conservation laws depend on the underlying viscous mechanism, namely different viscosity operators lead to different limit solutions. Standard numerical schemes for approximating conservation laws do not take into account this fact and converge to solutions that are not necessarily physically relevant. We design numerical schemes that incorporate explicit information about the underlying viscosity mechanism and approximate the physical-viscosity solution. Numerical experiments illustrating the robust performance of these schemes are presented.

Additive schemes (splitting schemes) for some systems of evolutionary equations

On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied problems the individual components of the vector of unknowns are coupled together and then splitting schemes are applied in order to get a simple problem for evaluating components at a new time level. Typically, the additive operator-difference schemes for systems of evolutionary equations are constructed for operators coupled in space. In this paper we investigate more general problems where coupling of derivatives in time for components of the solution vector takes place. Splitting schemes are developed using an additive representation for both the primary operator of the problem and the operator at the time derivative. Splitting schemes are based on a triangular two-component representation of the operators.

Solution blowup for systems of shallow-water equations

We consider initial-boundary value problems for systems of shallow-water equations. Using the testfunction method proposed by Pokhozhaev and Mitidieri, we study the effects of the boundary values and initial conditions on the occurrence, duration, and rate of blowup of the solutions of these problems. Under natural boundary conditions, we prove the existence of blowup in one- and two-dimensional problems in bounded and unbounded regions with dissipation and dispersion.

Solving initial-boundary value problems for systems of partial differential equations using neural networks and optimization techniques

A general system of the time-dependent partial differential equations containing several arbitrary initial and boundary conditions is considered. A hybrid method based on artificial neural networks, minimization techniques and collocation methods is proposed to determine a related approximate solution in a closed analytical form. The optimal values for the corresponding adjustable parameters are calculated. An accurate approximate solution is obtained, that works well for interior and exterior points of the original domain. Numerical efficiency and accuracy of the hybrid method are investigated by two-test problems including an initial value and a boundary value problem for the two-dimensional biharmonic equation.

Boundary Conditions for Coupled Quasilinear Wave Equations with Application to Isolated Systems

We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form [0, T] × Σ, where Σ is a compact manifold with smooth boundaries ∂Σ. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on ∂Σ. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell’s equations in the Lorentz gauge and Einstein’s gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.

A priori estimates for solutions of the first initial boundary-value problem for systems of fully nonlinear partial differential equations

We prove a priori estimates for a solution of the first initial boundary-value problem for a system of fully nonlinear partial differential equations (PDE) in a bounded domain. In the proof, we reduce the initial boundary-value problem to a problem on a manifold without boundary and then reduce the resulting system on the manifold to a scalar equation on the total space of the corresponding bundle over the manifold.

Initial-Boundary Value Problems for Conservation Laws

We establish some general concepts concerning boundary conditions for systems of conservation laws defined on one-dimensional space intervals, and prove some general results. Among these is an extension of Glimm's existence theorem to initial-boundary value problems. We also prove the convergence of the time averages of the solutions to certain initial-boundary value problems for systems arising in fluid flows in porous media, isentropic gas dynamics, and elasticity theory.

Measure-Valued Solutions to Initial-Boundary Value Problems for Certain Systems of Conservation Laws: Existence and Dynamics

A framework for studying initial-boundary value problems for systems of conservation laws, in what concerns to the existence of measure-valued solutions and their asymptotic behavior, is developed here with the helpful introduction of a class of flux maps which allow a rather complete treatment of these questions including systems of practical importance as those arising in multiphase flow in porous media. The systems of this class may, in general, admit umbilic points, submanifolds where genuine nonlinearity fails, as well as elliptic regions. We prove the existence of measure-valued solutions by using the vanishing viscosity method and, also, finite difference schemes. The main result about the dynamics of the measure@-valued solutions is that for certain special boundary values, given by constant states, the time-averages of these m@-v solutions converge weakly to the Dirac measure concentrated at those states, for a.e. space variable. The rate of convergence of the time-averages of the expected values can be estimated by properties of the flux maps only.

Reduction of the basic initial-boundary value problems for the Navier-Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations

We consider initial-boundary value problems for systems of Navier-Stokes equations, prescribing on the boundary the velocities, the stresses or the normal component of the velocity and the tangential stresses. We show that they can be reduced to initial-boundary value problems for systems of the form vt + Av + Kv=f, where A is a linear elliptic operator, containing a nonlocal term, while K is a nonlinear operator. For these problems we prove a local solvability theorem in the Sobolev-Slobodetskii spaces W2l, l/2.

An adaptive moving grid method for one-dimensional systems of partial differential equations

We describe a fully adaptive, moving grid method for solving initial-boundary value problems for systems of one-space dimensional partial differential equations whose solutions exhibit rapid variations in space and time. The method, based on finite-differences, is of the Lagrangian type and has been derived through a co-ordinate transformation which leads to equidistribution in space of the second derivative. Our technique is “intermediate” between static regridding methods, where nodes remain fixed for intervals of time, and continuously moving grid methods, where the node movement and the PDE integration are fully coupled. In our approach, the computation of the moving grids and the solution on these grids are carried out separately, while the nodes are moved at each time-step. Two error monitors have been implemented, one to govern the time-step selection and the other to eventually adapt the number of moving nodes. The method allows the use of different moving grids for different components in the PDE system. Numerical experiments are presented for a set of five sample problems from the literature, including two problems from combustion.

Asymptotics for a system of nonlinearly coupled wave equations with an application to the galloping oscillations of overhead transmission lines

In this paper an asymptotic theory for a class of initial-boundary value problems for systems of weakly and nonlinearly coupled wave equations is presented. The theory implies the well-posedness of the problem in the classical sense and the asymptotic validity of formal approximations on long time scales.