Vol 78, No 4 (2023)

Year: 2023
Articles: 8
URL: https://bakhtiniada.ru/0042-1316/issue/view/7534

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Averaging and mixing for stochastic perturbations of linear conservative systems

Huang G., Kuksin S.B.

Abstract

We study stochastic perturbations of linear systems of the form\begin{equation}dv(t)+Av(t) dt =\varepsilon P(v(t)) dt+\sqrt{\varepsilon} \mathcal{B}(v(t)) dW (t), \qquad v\in\mathbb{R}^D, \tag{*}\end{equation}where $A$ is a linear operator with non-zero imaginary spectrum.It is assumed that the vector field $P(v)$and the matrix function $\mathcal{B}(v)$ are locally Lipschitz with at most polynomial growth at infinity, that the equationis well-posed and a few of first moments of the norms of solutions $v(t)$ are bounded uniformly in $\varepsilon$. We use Khasminski'sapproach to stochastic averaging to show that, as $\varepsilon\to0$, a solution $v(t)$, written in the interaction representation interms of the operator $A$, for $0\leqslant t\leqslantConst\cdot\varepsilon^{-1}$ converges in distribution to a solution of an effective equation.The latter is obtained from $(*)$ by means of certain averaging. Assuming that equation $(*)$ and/or the effectiveequation are mixing, we examine this convergence further.Bibliography: 27 titles.

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Uspekhi Matematicheskikh Nauk. 2023;78(4):3-52

3-52

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Attractors. Then and now

Zelik S.V.

Abstract

This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishikand is based on a number of mini-courses taught by the author at the Universityof Surrey (UK) and Lanzhou University (China). It discusses the classicaland modern results of the theory of attractors for dissipative PDEs,including attractors for autonomous and non-autonomous equations,dynamical systems in general topological spaces, various types of trajectory,pullback and random attractors, exponential attractors,determining functionals and inertial manifolds, as well as the dimension theoryfor the classes of attractors mentioned above. The theoretical resultsare illustrated by a number of clarifying examples and counterexamples.Bibliography: 248 titles.

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Uspekhi Matematicheskikh Nauk. 2023;78(4):53-198

53-198