Attractors of periodic processes and estimates of their dimension
type: Item
instance of: scholarly article
title: Attractors of periodic processes and estimates of their dimension
zbMATH Open document ID: 0841.35016
DOI: 10.1007/BF02309145
published in: Mathematical Notes
publication date: 1996-02-21T00:00:00Z
review text: The authors consider abstract nonlinear periodic evolution equations \[ \partial_t u= A(u, t)\quad\text{for}\quad t\geq \tau,\quad u|_{t= \tau}= u_\tau\tag{1} \] on a Banach space \(E\) (\(A(u, t+ p)= A(u, t)\), where \(p> 0\) is the period) and the corresponding to (1) process \(U(t, \tau)\). Using the semigroup \(S(t)\) on the extended phase space \(E\times \mathbb{T}^1\) (\(\mathbb{T}^1\) is the circle \(\mathbb{R}\pmod p\)), which is defined by \[ S(t)(u, \sigma):= (U(t+ \sigma, \sigma)u,\;(t+ \sigma)(\text{mod }p)), \] attractors (i.e. minimal compact attracting sets) for (1) are constructed and their Hausdorff and fractal dimensions are estimated. Further, applications are given to the two-dimensional Navier-Stokes system with periodic external forces as well as to reaction diffusion systems and to dissipative hyperbolic equations with periodic terms.
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35Q30
zbMATH DE Number: 846475
zbMATH Keywords: periodic processes
zbMATH Keywords: abstract evolution equations
zbMATH Keywords: periodic terms
zbMATH Keywords: attractors
zbMATH Keywords: Hausdorff and fractal dimensions
zbMATH Keywords: two-dimensional Navier-Stokes system
zbMATH Keywords: reaction diffusion systems
zbMATH Keywords: dissipative hyperbolic equations
author: Mark Iosifovich Višik
author: Vladimir V. Chepyzhov
MaRDI profile type: Publication
cites work: Q4313959
cites work: Q3805109
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Attractors for damped nonlinear hyperbolic equations
cites work: Attractors of asymptotically compact processes and applications to nonlinear partial differential equations
cites work: Q3997729
cites work: Attractors of non-autonomous dynamical systems and their dimension
cites work: Nonautonomous evolution equations and their attractors
cites work: Q4285039
cites work: Q5618254
cites work: Upper and lower estimates for the dimension of attractors of partial differential evolution equations
cites work: Q3322454
cites work: Q4129141
cites work: Q5584121
full work available at URL: https://doi.org/10.1007/bf02309145
OpenAlex ID: W2081357154
description: scientific article; zbMATH DE number 846475