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 < p, q < +∞, let
Then the embedding of W into Lp([0, T]; X) is also compact