# Monthly Archives: April 2018

## Aubin-Lions Lemma

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.