Aubin–Lions lemma

In mathematics, the Aubin–Lions lemma (or theorem) is a result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove the existence of solutions one first constructs approximate solutions (for example, by a Galerkin method or by mollification of the equation), then uses the compactness lemma to show that there is a convergent subsequence of approximate solutions whose limit is a solution.

The result is named after the French mathematicians Thierry Aubin and Jacques-Louis Lions. In the original proof by Aubin,[1] the spaces X0 and X1 in the statement of the lemma were assumed to be reflexive, but this assumption was removed by Simon,[2] so the result is also referred to as the Aubin–Lions–Simon lemma.[3]

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. For 1 ≤ p, q ≤ +∞, let

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

(i) If p  < +∞, then the embedding of W into Lp([0, T]; X) is compact.

(ii) If p  = +∞ and q  >  1, then the embedding of W into C([0, T]; X) is compact.

Notes

References