Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations
type: Item
instance of: scholarly article
title: Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations
zbMATH Open document ID: 1415.35059
author: Caidi Zhao
author: Gang Xue
author: Grzegorz Łukaszewicz
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2018-12-27T00:00:00Z
review text: The authors give a necessary and sufficient condition for the existence of pullback-$\mathcal{D}$ attractors for processes defined on a Banach space of infinite sequences. The result generalizes the results of \textit{X.-Q. Zhao} and \textit{S. Zhou} [ibid. 9, No. 3--4, 763--785 (2008; Zbl 1149.37040)]. The setup is suitable for analyzing lattice dynamical systems. An application of the main result is given to a discrete system associated with the non-autonomous Klein-Gordon-Schrödinger system, an important set of partial differential equations in quantum field theory. The discrete system studied here was also examined by \textit{C. Zhao} and \textit{S. Zhou} [J. Math. Anal. Appl. 332, No. 1, 32--56 (2007; Zbl 1113.37057)] whereby, under certain assumptions on the non-autonomous source terms, those authors prove compact kernel sections for the discrete equations (pullback-$\mathcal{D}$ attractors have a more general basin of attraction than kernel sections). Based on the work in [\textit{G. Łukaszewicz} and \textit{J. C. Robinson}, Discrete Contin. Dyn. Syst. 34, No. 10, 4211--4222 (2014; Zbl 1351.37076)], the final result shows that the pullback-$\mathcal{D}$ attractor associated with the discretized system admits a family of invariant Borel probability measures. Such measures are useful in modeling turbulence.
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 76F20
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 34A33
Mathematics Subject Classification ID: 81Q05
zbMATH DE Number: 6996864
zbMATH Keywords: non-autonomous lattice system
zbMATH Keywords: invariant measures
zbMATH Keywords: Banach space if infinite sequences
reviewed by: Joseph Shomberg
MaRDI profile type: Publication
full work available at URL: dcdsb.2018122
OpenAlex ID: W2797402239
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DOI: 10.3934/DCDSB.2018122
description: scientific article; zbMATH DE number 6996864