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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 compactness criterion that is very useful in the study of nonlinear evolutionary partial differential equations. The result is named after the French mathematicians Thierry Aubin and Jacques-Louis Lions.

Statement of the lemma[edit]

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 < pq < +∞, let

W = \{ u \in L^p ([0, T]; X_0) | \dot{u} \in L^q ([0, T]; X_1) \}.

Then the embedding of W into Lp([0, T]; X) is also compact

References[edit]

  • Aubin, Jean-Pierre (1963). "Un théorème de compacité. (French)". C. R. Acad. Sci. Paris 256. pp. 5042–5044. MR 0152860. 
  • 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,. 

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