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

• Yes, this is too vague. Dec 24, 2013 at 1:48
• May be I should ask, are there any famous paper relate to Aubin-Lions Lemma after Simon's? Dec 24, 2013 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? Dec 25, 2013 at 2:18
• This revision is a much more appropriate question. Dec 26, 2013 at 17:53
• 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)
– user45757
Jan 20, 2014 at 7:55

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 ...

• 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. Feb 25, 2014 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 .... Feb 26, 2014 at 17:44