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I have a question about Aubin-Lions Lemma, the standard Aubin-Lions lemma need those Banach Space be reflexive spaces, are there any version of Aubin-Lions without reflexivity?

Standard aubin-lions:

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Yes, this is too vague. – Deane Yang Dec 24 '13 at 1:48
May be I should ask, are there any famous paper relate to Aubin-Lions Lemma after Simon's? – user44565 Dec 24 '13 at 1:56
@Deane Yang I have edited my question, now it is clearly. I main concern that can we remove the reflexivity in some special cases? – user44565 Dec 25 '13 at 2:18
This revision is a much more appropriate question. – Deane Yang Dec 26 '13 at 17:53
there are some new papers about aubin-lions-dubinskii lemma: [1] a note on aubin-lions-dubinskii lemmas( – user45757 Jan 20 '14 at 7:55
up vote 3 down vote accepted

I was wondering the same recently, and it seems to my that the answer is yes (you can get rid of reflexivity). Look at the paper of Jacques Simon : Compact sets in the spaces $L^p(0,T,B)$.

The paper claims to give sharp results in any regard and as far as I can see it only asks the spaces to be banach he gives for example Corollary 4 :

if $\{F\}$ is bounded in $L^q(0,T,X), \{F^\prime\}$ bounded in $ L^1(0,T,Y),$ with the usual assumption :$$X\underset{compact}{\hookrightarrow} B\underset{continous}{\hookrightarrow}Y,$$ then $\{F\}$ is relatively compact in $L^p(0,T,B)$, for $p<q$ where $X,B,Y$ are only Banach (assumption 8.1 in the paper). The corresponding result holds for $\{F\}\subset L^\infty$ and $\{F^\prime\}\subset L^r$ with $r>1$ (gives relative compactness in $\mathcal{C}(0,T,B)$).

I guess this is why it is sometimes mentionned as Aubin-Lions-Simon's lemma ...

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By the way this a particuliar case of how the lemma is recalled in the paper mentionned in his comment by user45757. So useless post I guess. But this is still the good reference for a proof. – user42070 Feb 25 '14 at 17:34
It makes me think that the article you mentionned on Wikipedia is really poor and misleading regarding the proper assumptions one might need. If anyone want to improve it .... – user42070 Feb 26 '14 at 17:44

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