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:http://en.wikipedia.org/wiki/Aubin-Lions_lemma
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$\begingroup$ Yes, this is too vague. $\endgroup$– Deane YangCommented Dec 24, 2013 at 1:48
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$\begingroup$ May be I should ask, are there any famous paper relate to Aubin-Lions Lemma after Simon's? $\endgroup$– user44565Commented Dec 24, 2013 at 1:56
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$\begingroup$ @Deane Yang I have edited my question, now it is clearly. I main concern that can we remove the reflexivity in some special cases? $\endgroup$– user44565Commented Dec 25, 2013 at 2:18
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$\begingroup$ This revision is a much more appropriate question. $\endgroup$– Deane YangCommented Dec 26, 2013 at 17:53
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1$\begingroup$ there are some new papers about aubin-lions-dubinskii lemma: [1] a note on aubin-lions-dubinskii lemmas(link.springer.com/article/10.1007/s10440-013-9858-8) $\endgroup$– user45757Commented Jan 20, 2014 at 7:55
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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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$\begingroup$ 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. $\endgroup$ Commented Feb 25, 2014 at 17:34
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$\begingroup$ 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 .... $\endgroup$ Commented Feb 26, 2014 at 17:44