In this article, we prove the celebrated compactness lemma which will be used to show the global existence of weak solution of Navier-Stokes equation.
Lemma 1. Let and be three Banach spaces with
Suppose is reflexive. Then for each , there is a constant such that
Proof. If the statement were not true, then there exists a number and a sequence in such that
Note for each . So we may assume . Since , we may assume that in for some . Moreover, since , it follows that in . But
for all . Hence, letting , we obtain
which is a contradiction.
Theorem (Aubin-Lions). Let and be three Banach spaces with
Suppose that are reflexive. Then for and , we have
Proof. Since and are reflexive, so are and . It suffices to show that if weakly both in and , then strongly in .
Observe that it suffices to show that strongly in . Indeed, if strongly in , then by Lemma 1, for each , we have
Since weakly in , is bounded in . Thus,
Since was arbitrary chosen, strongly in .
Since , is bounded in . Hence, to prove that
we will show that
Then the conclusion follows from the dominated convergence theorem.
Let be fixed. Since is continuous on , it is uniformly continuous and thus it is absolutely continuous. So for all ,
Integrating this over , we also have
and so integration by part gives
for all .
By (1) and Hölder’s inequality, we obtain
Hence given , we can choose such that
For each , define
and let . Then
Since and weakly in we have
Since , . So weakly in . Since , strongly in . Thus,
This completes the proof.