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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, the spaces X0 and X1 in the statement of the lemma were assumed to be reflexive, but this assumption was removed by Simon, so the result is also referred to as the Aubin–Lions–Simon lemma.

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

$$W = \{ u \in L^p ([0, T]; X_0) | \dot{u} \in L^q ([0, T]; X_1) \}.$$(ii) If p = +∞ and q > 1, then the embedding of W into C([0, T]; X) is compact.

(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.

Notes

Aubin (1963)
Simon (1986)

Boyer & Fabrie (2013)

References

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 Lp(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,.