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 . By Banach-Alalogu theorem, there exists a subsequence of and we still denote it by the same indices. Since , we have in for some . Moreover, since
, it follows that in
for all . Hence, letting , we obtain
which is a contradiction. This completes the proof.
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
Then note that weakly in . Indeed, let . Then we have
Since weakly in and can be regarded as a function in , the right hand side converges to .
Thus, converges to 0 strongly in . Hence,
This completes the proof.