Vladimir Victorovich Chepyzhov

We consider the reaction–diffusion system of equations with rapidly oscillating terms in the equation and in boundary conditions in a domain with locally periodic oscillating boundary. In the subcritical case (the Fourier boundary condition is changed to the Neumann boundary condition in the limit) we proved that the trajectory attractors of this system converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction–diffusion systems in domain independent of the small parameter, characterizing the oscillation rate. To obtain the results we use the approach of homogenization theory, asymptotic analysis and methods of the theory concerning trajectory attractors of evolution equations. Defining the appropriate functional and topological spaces with weak topology, we prove the existence of trajectory attractors and global attractors for these systems. Then we formulate the main Theorem and prove it with the help of auxiliary Lemmata. Applying the homogenization method and asymptotic analysis we derive the homogenized (limit) system of equations, prove the existence of trajectory attractors and global attractors and show the convergence of trajectory and global attractors.

In this paper, we consider the homogenization problem in a micro inhomogeneous domain with a rapidly oscillating boundary. It is assumed that a system of nonlinear reaction--diffusion equations with rapidly oscillating terms and dissipation is considered in the domain. On the locally periodic oscillating part of the boundary, the third boundary condition with rapidly oscillating coefficients and a small parameter characterizing the oscillation of the boundary to some degree is imposed. Depending on the degree of the small parameter in the boundary condition,various homogenized (limit) problems are obtained and the convergence of the trajectory attractors of the givensystem to the attractors of the homogenized system is proved. Critical, subcritical and supercritical cases of attractor behavior as the small parameter tends to zero are carefully studied. The paper also considers problems in a domain with a random rapidly oscillating boundary. In this case, a homogenized system of reaction--diffusion equations with deterministic coefficients is obtained in the case of a statistically homogeneous random structure of the boundary. A theorem on the convergence of random trajectory attractors of the initial given system of reaction-diffusion equations to a deterministic attractor of the homogenized (limit) system of reaction--diffusion equations is also proved. The paper also proves the convergence of global attractors in the case of uniqueness of solutions, which in turn is proved for nonlinearity in a system of equations of a special type.

This paper is devoted to studying the reaction - diffusion systems with rapidly oscillating coefficients in the equations and in boundary conditions in domains with locally periodic oscillating boundary; on this boundary a Robin boundary condition is imposed. We consider the supercritical case, when the homogenization changes the Robin boundary condition on the oscillating boundary is to the homogeneous Dirichlet boundary condition in the limit as the small parameter, which characterizes oscillations of the boundary, tends to zero. In this case, we prove that the trajectory attractors of these systems converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction - diffusion systems in the domain independent of the small parameter. For this aim we use the homogenization theory, asymptotic analysis and the approach of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of dissipative evolution equations. The homogenization method and asymptotic analysis are used to derive the homogenized reaction - diffusion system and to prove the convergence of solutions. First we define the appropriate auxiliary functional spaces with weak topology, then, we prove the existence of trajectory attractors for these systems and formulate the main Theorem. Finally, we prove the main convergence result with the help of auxiliary lemmas.

The purpose of this work is to study the limit behaviour of trajectory attractors for some equations and systems from mathematical physics depending on a small parameter when this small parameter approaches zero. The main attention is given to the cases when, for the limit equation, the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. The following problems are considered: approximation of the 3D Navier–Stokes system using the Leray α-model, homogenization of the complex Ginzburg–Landau equation in a domain with dense perforation, and zero viscosity limit of 2D Navier–Stokes system with Ekman friction. Methods. In this paper, the method of trajectory dynamical systems and trajectory attractors is used that is especially effective in the study of complicated partial differential equations for which the uniqueness theorem for a solution of the corresponding initial-value problem does not hold or is not proved. Results. For all problems under the consideration, we obtain the limit equations and prove the Hausdorff convergence for trajectory attractors of the initial equations to the trajectory attractors of the limit equations in the appropriate topology when the small parameter tends to zero. Conclusion. In the work, we demonstrate that the method of trajectory attractors is highly effective in the study of dissipative equations of mathematical physics with small parameter. We succeed to find the limit equations and to prove the convergence of trajectory attractors of the considered equations to the trajectory attractors of the limit (homogenized) equations in the corresponding topology as small parameter is vanishes.

The Ginzburg-Landau equation with rapidly oscillating terms in the equation and boundary conditions in a perforated domain was considered. Proof was given that the trajectory attractors of this equation converge weakly to the trajectory attractors of the homogenized Ginzburg-Landau equation. To do this, we use the approach from the articles and monographs of V.V. Chepyzhov and M.I. Vishik about trajectory attractors of evolutionary equations, and we also use homogenization methods that appeared at the end of the 20th century. First, we use asymptotic methods to construct asymptotics formally, and then we justify the form of the main terms of the asymptotic series using functional analysis and integral estimates. By defining the corresponding auxiliary function spaces with weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Then, we formulate the main theorems and prove them by using auxiliary lemmas. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized Ginzburg-Landau equation in the subcritical case, and they disappear in the supercritical case.

A reaction-diffusion system in a domain with randomly located obstacles was considered. When studying the problem, we sat the homogeneous Dirichlet condition on the outer boundary of the domain and the Neumann condition on the boundary of the cavities. Under such assumptions, it was proven that random trajectory attractors of this system with random coefficients converge in some weak topology to the deterministic trajectory attractor of a homogenized reaction-diffusion system with deterministic coefficients in a homogeneous domain without obstacles. In the case of uniqueness, we obtained weak convergence of random global attractors to a deterministic global attractor.Bibliography: 19 titles.

In the paper we consider a problem for complex Ginzburg–Landau equations in a medium with locally periodic small obstacles. It is assumed that the obstacle surface can have different conductivity coefficients. We prove that the trajectory attractors of this system converge in a certain weak topology to the trajectory attractors of the homogenized Ginzburg–Landau equations with an additional potential (in the critical case), without an additional potential (in the subcritical case) in the medium without obstacles, or disappear (in the supercritical case).