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> 



About Aubin-Lions Lemma
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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=Simon&s5=Compact%20sets&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) : 
*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 ...


