TITLE: About Aubin-Lions Lemma
QUESTION [5 upvotes]: 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

REPLY [4 votes]: 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 ...