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

*X*_{0} *X* and

*X*_{1} be three Banach spaces with

*X*_{0} ⊆

*X* ⊆

*X*_{1} Suppose that

*X*_{0} is compactly embedded in

*X* and that

*X* is continuously embedded in

*X*_{1}; suppose also that

*X*_{0} and

*X*_{1} are reflexive spaces For 1 <

*p* *q* < +∞ let

- $W\; =\; \{\; u\; in\; L^\{p\}\; (T;\; X\_\{0\})\; |\; dot\{u\}\; in\; L^\{q\}\; (T;\; X\_\{1\})\; \}$

Then the

embedding of

*W* into

*L*^{p}(

[1];

*X*) is also compact

## References