Finite topological space

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

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

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

References

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