An alternative proof of the Aubin-Lions lemma
type: Item
instance of: scholarly article
title: An alternative proof of the Aubin-Lions lemma
zbMATH Open document ID: 1287.46033
author: Rafael Serrano
published in: Archiv der Mathematik
publication date: 2013-11-12T00:00:00Z
review text: \noindent The author uses a generalized version of the Weyl-Riesz compactness criterion of subsets of Lebesgue-Bochner spaces (due to Gutman) to give a new proof of the Aubin-Lions theorem. The Aubin-Lions theorem is: Let \(X_0, X_1, X\) be Banach spaces satisfying \(X_0 \subset X \subset X_1\) with continuous embeddings. Let \(T>0\) be finite, and let \(\alpha_0, \alpha_1 \in [1, \infty)\). Consider the space \[ \mathcal{Y} := \{ v \in L^{\alpha_0}(0, T; X_0): v^{\prime} \in L^{\alpha_1}(0, T; X_1) \} \] which is a Banach space with the obvious norm. If the embedding \(X_0 \hookrightarrow X_1\) is compact, and if \(\alpha_0\), \(\alpha_1\) are such that if \(\alpha_1 = 1\), then \(\alpha_0 = 1\), the embedding \(\mathcal{Y} \hookrightarrow L^{\alpha_0}(0, T; X)\) is also compact. The Aubin-Lions theorem is used in proofs of the existence of solutions of nonlinear evolution equations.
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B50
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 6226073
zbMATH Keywords: compactness
zbMATH Keywords: Aubin-Lions theorem
zbMATH Keywords: Lebesgue-Bochner spaces
zbMATH Keywords: non-reflexive Banach spaces
reviewed by: Raymond Johnson
MaRDI profile type: Publication
full work available at URL: s00013-013-0552-x
OpenAlex ID: W2008894288
Wikidata QID: Q124792347
cites work: Q5591023
cites work: Compact Perturbations of m-Accretive Operators in General Banach Spaces
cites work: Nonlinear partial differential equations with applications
cites work: Compact sets in the space \(L^ p(0,T;B)\)
cites work: Q4828519
DOI: 10.1007/S00013-013-0552-X
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description: scientific article; zbMATH DE number 6226073