An optimal compactness theorem and application to elliptic-parabolic systems
type: Item
instance of: scholarly article
title: An optimal compactness theorem and application to elliptic-parabolic systems
zbMATH Open document ID: 1001.46049
DOI: 10.1016/S0893-9659(00)00153-1
author: J.-M. Rakotoson
author: R. Temam
published in: Applied Mathematics Letters
publication date: 2001-08-07T00:00:00Z
reviewed by: Rudolf Kodnár
Mathematics Subject Classification ID: 46N20
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 1631135
zbMATH Keywords: compactness
zbMATH Keywords: elliptic-parabolic systems
zbMATH Keywords: Aubin-Lions theorem
MaRDI profile type: Publication
cites work: Q5890172
cites work: Q3684480
cites work: Q4841248
cites work: Q4178676
cites work: Infinite-dimensional dynamical systems in mechanics and physics
Wikidata QID: Q127846897
review text: The authors prove and apply the following theorem: Let \((V,\|\cdot \|)\), \((H,|\cdot |)\) be two separable Hilbert spaces. Assume that \(V\subset H\) with a compact and dense imbedding. Consider a sequence \(\{ u^\varepsilon\}_{\varepsilon >0}\) converging weakly to function \(u\) in \(L^2 (0, T;V)\), \(T<+ \infty\). Then \(u^\varepsilon\) converges strongly to \(u\) in \(L^2 (0, T;H)\) if and only ifNEWLINENEWLINENEWLINE1) \(u^\varepsilon (t)\) converges to \(u(t)\) weakly in \(H\) for a.e. \(t\),NEWLINENEWLINENEWLINE2) \(\lim_{\text{meas}(E) \to 0,E<\langle 0,T\rangle} \sup_{\varepsilon> 0}\int_E\bigl |u^\varepsilon (t)\bigr |^2dt=0.\)
description: scientific article; zbMATH DE number 1631135