On relation between attractors for single and multivalued semiflows for a certain class of PDEs
type: Item
instance of: scholarly article
title: On relation between attractors for single and multivalued semiflows for a certain class of PDEs
zbMATH Open document ID: 1420.35049
author: Piotr Kalita
author: Grzegorz Łukaszewicz
author: Jakub Siemianowski
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2019-03-21T00:00:00Z
review text: This article is about dissipative PDE, which admit weak solutions that are \textit{not} necessarily unique, and a global attractor. Such problems give rise to multivalued flows (e.g. generalized semiflows [\textit{J. M. Ball}, J. Nonlinear Sci. 7, No. 5, 475--502 (1997; Zbl 0903.58020); erratum ibid. 8, No. 2, 233 (1998); Corrected version appears in: Mechanics: from theory to computation. Essays in honor of Juan-Carlos Simo. Papers invited by Journal of Nonlinear Science editors. New York, NY: Springer. 447--474 (2000; Zbl 0958.35101)] or trajectory attractors [\textit{V. V. Chepyzhov} and \textit{M. I. Vishik}, C. R. Acad. Sci., Paris, Sér. I 321, No. 10, 1309--1314 (1995; Zbl 0843.35038); ibid. 321, No. 2, 153--158 (1995; Zbl 0837.35059)]). For the class of problems in which these weak solutions may be bootstrapped into existing (unique) strong solutions, it is shown that the global attractor for the strong solutions coincides with global attractors for weak solutions. One illuminating aspect to this is that the finite-dimensionality of the global attractor, when this is present for the unique strong solutions, carries over for the case of the weak solutions. Two applications are given.
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 76D03
Mathematics Subject Classification ID: 80M35
zbMATH DE Number: 7039946
zbMATH Keywords: m-semiflow
zbMATH Keywords: global attractor
zbMATH Keywords: fractal dimension
zbMATH Keywords: thermomicropolar fluid
zbMATH Keywords: quasigeostrophic equation
zbMATH Keywords: trajectory attractors
reviewed by: Joseph Shomberg
MaRDI profile type: Publication
full work available at URL: dcdsb.2019012
OpenAlex ID: W2907962505
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Wikidata QID: Q128567344
DOI: 10.3934/DCDSB.2019012
description: scientific article; zbMATH DE number 7039946