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,^[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 ${\displaystyle 1\leq p,q\leq \infty }$, let

${\displaystyle W=\{u\in L^{p}([0,T];X_{0})\mid {\dot {u}}\in L^{q}([0,T];X_{1})\}.}$

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

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

See also

• Lions–Magenes lemma