Aubin–Lions lemma
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In mathematics, the Aubin–Lions lemma (or theorem) is the 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 Jean-Pierre 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]