An initial boundary-value problem for a system of equations of magnetohydrodynamics

In the present paper we consider one initial boundary-value problem for a system of equations of magnetohydrodynamics in the case where it is necessary to take into account the displacement currents in the Maxwell system of equations. We prove a local (in time) unique solvability of this problem in the Sobolev spaces. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 228–254, April–June, 2000. Translated by R. Lapinskas     

An initial boundary-value problem for model electromagnetoelasticity system

In this paper we consider an initial boundary-value problem related to the electrodynamics of vibrating elastic media. The aim is to prove an existence and uniqueness result for a model describing the nonlinear interactions of the electromagnetic and elastic waves. We assume that the motion of the continuum occurs at velocities that are much smaller than the propagation velocity of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations, one of them is the hyperbolic equation (an analog of the Lamé system) and another one is the parabolic equation (an analog of the diffusion Maxwell system). One stability result is proved too.     

Global solutions of a two-dimensional initial boundary-value problem for a system of semilinear magnetoelasticity equations

We prove the theorem on the existence and uniqueness of global solutions of a system of semilinear magnetoelasticity equations in a two-dimensional space.     

On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.     

On the two-point boundary-value problem for equations of geodesics

For equations of “geodesic spray” type with continuous coefficients on a complete Riemannian manifold, some interrelations between certain geometric characteristics, the distance between points, and the norm of the right-hand side that guarantee the solvability of the boundary-value problem are found. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 65–70, 2005.     

A dynamic boundary-value problem without initial conditions for almost linear parabolic equations

We study a dynamic boundary-value problem without initial conditions for linear and almost linear parabolic equations. First, we establish conditions for the existence of a unique solution of a problem without initial conditions for a certain abstract implicit evolution equation in the class of functions with exponential behavior as t → −∞. Then, using these results, we prove the existence of a unique solution of the original problem in the class of functions with exponential behavior at infinity.     

The solvability of the second initial boundary-value problem for the linear,time-dependent system of Navier-Stokes equations

In a class of functions with Holder-continuous derivatives unique solvability is is proved for the problem of determining a solution of the linear, time-dependent system of Navier-Stokes equations with boundary data , where are the direction cosines of the exterior normal to the boundary and are the components of the stress tensor.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 200–218, 1977.     

On the boundary-value problem for a class of equations of mixed type in an unbounded domain

In this paper, we study the boundary-value problem for an equation of mixed type with singular coefficient. The uniqueness of the solution of the problem is proved using the extremum principle and the existence of a solution to the problem is established by the method of integral equations.     

Nonlocal boundary-value problem for parabolic equations

We study the problem for Shilov parabolic equations of arbitrary order with constant coefficients with conditions nonlocal in time and periodic in space variables. We establish conditions for the existence and uniqueness of a classical solution of the problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1621–1626, December, 1994.The work was supported by the Foundation for Fundamental Studies of Ukrainian State Committee on Science and Technology.     

First boundary-value problem for a class of integrodifferential equations

Lithuanian Mathematical Journal -     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

On the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient

We study the initial boundary-value problem for a nonlinear Sobolev-type equation with variable coefficient. We obtain sufficient conditions for both global and local (in time) solvability. In the case of local (but not global) solvability, we obtain upper and lower bounds for the existence time of the solution in the form of explicit and quadrature formulas.