In mathematics, the **Aubin–Lions lemma** (or theorem) is a 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 Thierry Aubin and Jacques-Louis Lions. In the original proof by Aubin,^{[1]} the spaces *X*_{0} and *X*_{1} 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]}

## Statement of the lemma[edit]

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}. For 1 ≤ *p*, *q* ≤ +∞, let

(i) If *p* < +∞, then the embedding of *W* into *L*^{p}([0, *T*]; *X*) is compact.

(ii) If *p* = +∞ and *q* > 1, then the embedding of *W* into *C*([0, *T*]; *X*) is compact.

## Notes[edit]

## References[edit]

- Aubin, Jean-Pierre (1963). "Un théorème de compacité. (French)".
*C. R. Acad. Sci. Paris***256**. pp. 5042–5044. MR 0152860. - Boyer, Franck; Fabrie, Pierre (2013).
*Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models*. Applied Mathematical Sciences 183. New York: Springer. pp. 102–106. ISBN 978-1-4614-5975-0. (Theorem II.5.16)

- Lions, J.L. (1969).
*Quelque methodes de résolution des problemes aux limites non linéaires*. Paris: Dunod-Gauth. Vill. MR 259693.

- Roubíček, T. (2013).
*Nonlinear Partial Differential Equations with Applications*(2nd ed.). Basel: Birkhäuser. ISBN 978-3-0348-0512-4. (Sect.7.3)

- 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. MR 1422252. (Proposition III.1.3)

- Simon, J. (1986). "Compact sets in the space L
^{p}(O,T;B)".*Annali di Matematica Pura ed Applicata***146**. pp. 65–96. MR 0916688 (89c:46055).

- X.Chen, A.Jungel and J.-G.Liu (2014). "A note on Aubin-Lions-Dubinskii lemmas".
*Acta Appl. Math. to appear,*.

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