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In mathematics, the Aubin–Lions lemma (or theorem) is the 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 Jean-Pierre Aubin and Jacques-Louis Lions. In the original proof by Aubin,[1] the spaces X0 and X1 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

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. For 1 ≤ pq ≤ +∞, let

${\displaystyle W=\{u\in L^{p}([0,T];X_{0})|{\dot {u}}\in L^{q}([0,T];X_{1})\}.}$

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

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

## References

• Aubin, Jean-Pierre (1963). "Un théorème de compacité. (French)". C. R. Acad. Sci. Paris 256. pp. 5042–5044. MR 0152860.
• Barrett, John W.; Süli, Endre (2012). "Reflections on Dubinskii's nonlinear compact embedding theorem". Publications de l'Institut Mathématique (Belgrade) (N.S.). 91 (105). pp. 95–110. MR 2963813.
• 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 Lp(O,T;B)". Annali di Matematica Pura ed Applicata 146. pp. 65–96. MR 0916688 (89c:46055).
• Chen, X.; Jüngel, A.; Liu, J.-G. (2014). "A note on Aubin-Lions-Dubinskii lemmas". Acta Appl. Math. 133. pp. 33–43. MR 3255076.

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