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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 compact space|compactness criterion that is very useful in the study of nonlinear evolutionary partial differential equations

    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; suppose also that X0 and X1 are reflexive spaces For 1 < p q < +∞ let
    W = { u in L^{p} ( T; X_{0}) | dot{u} in L^{q} ( T; X_{1}) }

    Then the embedding of W into Lp([1]X) is also compact

    References

    • (Theorem III13)