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

. But

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 .

Suppose thus that weakly both in and . Since weakly in , is bounded in . Set

(1)

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.