Aubin-Lions lemma

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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.

[edit] 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

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

[edit] 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.  MR1422252 (Theorem III.1.3)