Английская Википедия:Aubin–Lions lemma

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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,[1] the spaces X0 and X1 in the statement of the lemma were assumed to be reflexive, but this assumption was removed by Simon,[2] so the result is also referred to as the Aubin–Lions–Simon lemma.[3]

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>1\leq p, q\leq\infty</math>, let

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

(i) If <math>p<\infty</math> then the embedding of Шаблон:Mvar into <math>L^p([0,T];X)</math> is compact.

(ii) If <math>p=\infty</math> and <math>q>1</math> then the embedding of Шаблон:Mvar into <math>C([0,T];X)</math> is compact.

See also

Notes

Шаблон:Reflist

References

Шаблон:Functional analysis

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  1. ↑ Шаблон:Harvtxt
  2. ↑ Шаблон:Harvtxt
  3. ↑ Шаблон:Harvtxt
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