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# Aubin-Lions lemma

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 compact space|compactness criterion that is very useful in the study of nonlinear evolutionary partial differential equations

## 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
$W = \left\{ u in L^\left\{p\right\} \left(T; X_\left\{0\right\}\right) | dot\left\{u\right\} in L^\left\{q\right\} \left(T; X_\left\{1\right\}\right) \right\}$

Then the embedding of W into Lp([1]X) is also compact

## References

• (Theorem III13)