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 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 $1\le p,q\le \mathrm{\infty }$, let

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

(i) If $p<\mathrm{\infty }$ then the embedding of Template:Mvar into $L^p([0,T];X)$ is compact.

(ii) If $p=\mathrm{\infty }$ and $q>1$ then the embedding of Template:Mvar into $C([0,T];X)$ is compact.

See also

• Lions–Magenes lemma

References

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