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 French mathematicians Jean-Pierre Aubin and Jacques-Louis Lions. In the original proof by Aubin, 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, so the result is also referred to as the Aubin–Lions–Simon lemma.

Statement of the lemma

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 $1\leq p, q\leq\infty$, let

:$W = \.$

(i) If $p<\infty$ then the embedding of into L^p( ,TX) is compact.

(ii) If $p=\infty$ and $q>1$ then the embedding of into C( ,TX) is compact.

See also

* Lions–Magenes lemma

Notes

References

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* (Theorem II.5.16)
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* (Sect.7.3)
* (Proposition III.1.3)
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{{DEFAULTSORT:Aubin-Lions lemma
Banach spaces
Theorems in functional analysis
Lemmas in mathematical analysis
Measure theory