Aubin–Lions lemma
# 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 $X_0$ and $X_1$ 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 $X_0,$ 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 $X_1.$ For $1\leq p,q\leq \infty$, let

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

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

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

