# Aubin–Lions lemma

In mathematics, the Aubin–Lions lemma is a result in the theory of Sobolev spaces of Banach space-valued functions. More precisely, it is a compactness criterion that is very useful in the study of nonlinear evolutionary partial differential equations. The result is named after the French mathematicians Thierry Aubin and Jacques-Louis Lions.

## Statement of the lemma

Let X0, X and X1 be three Banach spaces with X0 ⊆ X ⊆ X1. Suppose that X0 is compactly embedded in X and that X is continuously embedded in X1; suppose also that X0 and X1 are reflexive spaces. For 1 < pq < +∞, let

$W = \{ u \in L^p ([0, T]; X_0) | \dot{u} \in L^q ([0, T]; X_1) \}.$

Then the embedding of W into Lp([0, T]; X) is also compact

## References

• Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 106. ISBN 0-8218-0500-2. MR 1422252. (Proposition III.1.3)