Aubin–Lions lemma

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 France mathematicians Jean-Pierre 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 [math]\displaystyle{ 1\leq p, q\leq\infty }[/math], let

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

(i) If [math]\displaystyle{ p\lt \infty }[/math] then the embedding of W into [math]\displaystyle{ L^p([0,T];X) }[/math] is compact.

(ii) If [math]\displaystyle{ p=\infty }[/math] and [math]\displaystyle{ q\gt 1 }[/math] then the embedding of W into [math]\displaystyle{ C([0,T];X) }[/math] is compact.

See also

• Lions–Magenes lemma

References

• Aubin, Jean-Pierre (1963). "Un théorème de compacité. (French)". C. R. Acad. Sci. Paris 256: pp. 5042–5044. 
• Barrett, John W.; Süli, Endre (2012). "Reflections on Dubinskii's nonlinear compact embedding theorem". Publications de l'Institut Mathématique (Belgrade). Nouvelle Série 91 (105): 95–110. doi:10.2298/PIM1205095B. 
• 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.. 
• 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. pp. 106. ISBN 0-8218-0500-2.  (Proposition III.1.3)
• Simon, J. (1986). "Compact sets in the space L^p(O,T;B)". Annali di Matematica Pura ed Applicata 146: 65–96. doi:10.1007/BF01762360. 
• Chen, X.; Jüngel, A.; Liu, J.-G. (2014). "A note on Aubin-Lions-Dubinskii lemmas". Acta Appl. Math. 133: pp. 33–43.