Homogenization of nonstationary Navier-Stokes equations in a domain with a grained boundary

Summary We prove the convergence of the homogenization process for a nonstationary Navier-Stokes system in a porous medium. The result of homogenization is Darcy's law, as in the case of the Stokes equation, but the convergence of pressures is in a different function space.This work was supported in part by INA-NAFTAPLIN, Geological Exploration and Development Division, Zagreb, Yugoslavia, and by SIZ I, Zagreb, Yugoslavia.     

Homogenization of a stochastically forced Hamilton-Jacobi equation

We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions, which are of independent interest.     

Homogenization of a nonlinear degenerate parabolic differential equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problem ɑtu∈-div(D(x/∈, u∈,▽u∈)+ K(x/∈, u∈))= f(x) with Dirichlet boundary condition. Here the operator D(y, s,s) is periodic in y and degenerated in ▽s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈→ 0 and also prove the corrector results of ▽u∈ to strong convergence.     

Analytical solution of the nonstationary boltzmann equation in the BGC model

Using the Case method, the analytical solution to the BGC nonstationary kinetic equation is constructed. As an application, the problem of a point-like source of heat or particles is considered. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 139–148, October, 1997.     

Homogenization of a boundary condition for the heat equation

An asymptotic analysis is given for the heat equation with mixed boundary conditions rapidly oscillating between Dirichlet and Neumann type. We try to present a general framework where deterministic homogenization methods can be applied to calculate the second term in the asymptotic expansion with respect to the small parameter characterizing the oscillations. Received August 20, 1999 / final version received March 1, 2000?Published online June 21, 2000     

Homogenization of the layered medium equation

A new partial differential equation to be called the layered medium equation is introduced, and it is proved that certain relevant initial or periodic boundary conditions give well-posed problems. Then, the homogenized limit of the layered medium equation is studied. It is shown to be preserved in limit in the limit in the physical problem in which the coefficients that arise from the dielectric layer are both proportional to thickness. Otherwise, a non-local problem is obtained as the limiting form     

Galerkin method for a nonstationary equation with a monotone operator

In the present paper, we consider the Galerkin method for a quasilinear differentialoperator equation with a leading self-adjoint operator A(t) and a subordinate monotone operator K. For the projection subspaces we take linear spans of eigenelements of an operator similar to the leading operator A(t). We obtain new estimates for the Galerkin method and consider applications to an initial-boundary value problem for a parabolic equation of higher order.     

Homogenization limit of a parabolic equation with nonlinear boundary conditions

Consider the parabolic equation

(E)     

Homogenization of nonlinear reaction-diffusion equation with a large reaction term

This paper deals with the homogenization of a second order parabolic operator with a large nonlinear potential and periodically oscillating coefficients of both spatial and temporal variables. Under a centering condition for the nonlinear zero-order term, we obtain the effective problem and prove a convergence result. The main feature of the homogenized equation is the appearance of a non-linear convection term.     

Homogenization of a parabolic model of ferromagnetism

This work deals with the homogenization of hysteresis-free processes in ferromagnetic composites. A degenerate, quasilinear, parabolic equation is derived by coupling the Maxwell-Ohm system without displacement current with a nonlinear constitutive law:

     

Problem of determining the nonstationary potential in a hyperbolic-type equation

We solve the problem of determining the hyperbolic equation coefficient depending on two variables. Some additional information is given by the trace of the direct problem solution on the hyperplane x = 0. We estimate the stability of the solution of the inverse problem under study and prove the uniqueness theorem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 220–225, August, 2008.     

A nonlinear integrodifferential equation of nonstationary transfer

Existence and uniqueness theorems for the solution of a mixed problem for a nonlinear nonstationary integrodifferential transfer equation of general form, as well as stability theorems for the solutions of this problem with respect to the perturbation of a parameter occurring in the equation and in the initial and boundary conditions, are established in the article.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 665–676, May, 1977.     

A retrial model in a nonstationary regime

Summary In this article we analyze a retrial queuing system where customers in the orbit join a queue with FCFS discipline. We adopt a nonstationary regime. We derive some probabilities using the theory of semiregenerative processes. We obtain an integral estimation for the difference between blocking probabilities in stationary and nonstationary regimes.     

Scattering data for the nonstationary Dirac equation

The existence of indeterminacy in the choice of scattering data for the auxiliary linear system for the Davey-Stewartson I-equation is noted. A connection is established between different scattering data and the corresponding conjugation matrix for the nonlocal Riemann problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 164, pp. 170–175, 1987.     

Integral equation method in three-dimensional problems of low-frequency electrodynamics

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostika, pp. 180–186, Izd. Moskovskogo Universiteta, Moscow, 1990.     

One case of analytic integrability of a nonstationary matrix ordinary differential equation

We find the general solution of the equation for the error in the matrix of direction cosines of the form \(\dot x\) = αx + β, where α(t) is a skew-symmetric matrix with zero main diagonal, β(t) is the product of a skew-symmetric matrix by a matrix exponential, and α(t) and β(t) are functions of direction cosines, the angular velocity of the moving trihedral, and the error in the determination of the angular velocity. We point out the possibility of using the general solution to construct approximate formulas for the analysis of the orientation error.     

Homogenization of a parabolic equation in perforated domain with Neumann boundary condition

In this article, we study the homogenization of the family of parabolic equations over periodically perforated domains