Aubin-Lions lemma

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.

tatement 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; suppose also that "X"₀ and "X"₁ are reflexive spaces. For 1 < "p", "q" < +∞, 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 "L"^"p"( [0, "T"] ; "X") is also compact

References

* cite book
last = Showalter
first = Ralph E.
title = Monotone operators in Banach space and nonlinear partial differential equations
series = Mathematical Surveys and Monographs 49
publisher = American Mathematical Society
location = Providence, RI
year = 1997
pages = p. 106
isbn = 0-8218-0500-2 MathSciNet|id=1422252 (Theorem III.1.3)