AP 22684340 "On the asymptotics of attractors of the Ginzburg-Landau complex equation in a perfected domain with an oscillating boundary"

The asymptotic analysis of problems in micro–homogeneous media (in particular, in porous media and perforated materials, composites) attracts the attention of specialists from various fields of science. Such problems have been studied in environments with periodic, almost periodic, locally periodic, as well as random (stochastic) microstructure. Problem statements in highly inhomogeneous porous media are of great interest, as noted above. The project is supposed to study the asymptotics of attractors for the Ginzburg-Landau equation in a new formulation (a highly inhomogeneous domain where the equation is considered). In many situations, the averaged equation changes (an additional term appears). Such effects have not been previously detected for the Ginzbhurg-Landau system of equations. The project assumes

1. To obtain averaged (limit) equations for the Ginzburg-Landau system of equations in cases of a boundary condition on an oscillating boundary with parameters satisfying the relations.
2. To prove the averaging theorems for the trajectory attractors of the complex Ginzburg-Landau equation in a perforated region with a rapidly oscillating outer boundary in the above cases.

The aim of the project is to study the asymptotic behavior of trajectory attractors for the complex Ginzburg-Landau equation in a perforated region with a rapidly oscillating outer boundary when the parameter characterizing the pore sizes and the distance between them, as well as the amplitude and frequency of the boundary oscillation, tends to zero.

Here we will only talk about the main results that will be obtained within the framework of the project. Naturally, we will talk about the results in a general form without a specific formulation of conditions and statements.

1. For the complex Ginzburg-Landau equation, in the case of a boundary condition on an oscillating boundary with parameters satisfying the relation a>b-1, an averaged problem will be obtained. Convergence in the weak topology of the attractors of the original problem to the attractor of the averaged problem will be proved.
2. For the complex Ginzburg-Landau equation, in the case of a boundary condition on an oscillating boundary with parameters satisfying the relation a=b-1, an averaged problem will be obtained. Convergence in the weak topology of the attractors of the original problem to the attractor of the averaged problem will be proved.
3. For the complex Ginzburg-Landau equation, in the case of a boundary condition on an oscillating boundary with parameters satisfying the relation aaveraged problem will be obtained. Convergence in the weak topology of the
attractors of the original problem to the attractor of the averaged problem will be proved.
4. Defense of a doctoral dissertation (PhD) in the specialty 8D05401-"Mathematics".
5. According to the results obtained, at least 2 (two) will be published within the framework of this project articles in journals from the first three quartiles by impact factor in the Web of Science database or having a CiteScore percentile in the Scopus database of at least 50. We publish an article in one of the following journals:
1. Mathematics (Q2) ;
2.Applicable Analysis and Discrete Mathematics (Q3);
3. Journal of Differential Equations (Q1);
4. Communications on Pure and Applied Mathematics (Q1);
5. Journal des Mathematiques Pures et Appliquees (Q1).