Homogenization of Attractors to Ginzburg-Landau Equations in Media with Locally Periodic Obstacles: Critical Case

Локальды периодты кеуектерi бар орталарда Гинсбург-Ландау теңдеулерiнiң аттракторларын орташалау: критикалық жағдай
Усреднение аттракторов уравнений Гинзбурга-Ландау в средах с локально периодическими препятствиями: критический случай
Bekmaganbetov K.A. Chechkin G.A. Chepyzhov V.V. Tolemis A.A.
2023 E.A. Buketov Karaganda University Publish house

Bulletin of the Karaganda University. Mathematics Series
2023 #111 Issue 3 11 - 27 pp.

In this paper the Ginzburg-Landau equation is considered in locally periodic porous medium, with rapidly oscillating terms in the equation and boundary conditions. It is proved that the trajectory attractors of this equation converge in a weak sense to the trajectory attractors of the limit Ginzburg-Landau equation with an additional potential term. For this aim we use an approach from the papers and monographs of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of evolution equations. Also we apply homogenization methods appeared at the end of the XX-th century. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) equation and prove the existence of trajectory attractors for this equation. Then we formulate the main theorem and prove it with the help of axillary lemmas.

attractors , Ginzburg-Landau equations , homogenization , nonlinear equations , perforated domain , porous medium , strange term , weak convergence

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M.V. Lomonosov Moscow State University, Kazakhstan Branch, Astana, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
M.V. Lomonosov Moscow State University, Moscow, Russian Federation
Institute of Mathematics with Computing Center Subdivision, the Ufa Federal Research Center of Russian Academy of Science, Ufa, Russian Federation
Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russian Federation
L.N. Gumilyov Eurasian National University, Astana, Kazakhstan

M.V. Lomonosov Moscow State University
Institute of Mathematics and Mathematical Modeling
M.V. Lomonosov Moscow State University
Institute of Mathematics with Computing Center Subdivision
Institute for Information Transmission Problems
L.N. Gumilyov Eurasian National University

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