 Aubin–Lions lemma

A useful version of the Aubin-Lions lemma concerns Banach spaces with embeddingswhere the first embedding is compact and the second continuous. For $1<p<\infty$ and $1<q\le \infty$, a family bounded in $L^p(0,T;X_0)$ whose distributional time derivatives are bounded in $L^q(0,T;X_1)$ is relatively compact in $L^p(0,T;X)$. The exponents and embedding hypotheses must be stated; an unspecified suitable derivative bound does not identify a version of the theorem.

For a bounded Lipschitz domain $\Omega$, Rellich-Kondrachov compactness theorem and continuous dual embeddings giveConsequently boundedness of $u_j$ in $L^2(0,T;H^1(\Omega ))$ and of ${\partial }_t{u}_j$ in $L^2(0,T;H^{-1}(\Omega ))$ yields a strongly convergent subsequence in $L^2(0,T;L^2(\Omega ))$. The spatial bound prevents fine-scale spatial oscillations, while the derivative bound controls time translations. This is why the lemma lets a Galerkin method pass to nonlinear terms, for which weak convergence alone is often insufficient. On an unbounded spatial domain, the displayed spatial compact embedding can fail through translations.

Aubin–Lions lemma is a result in the field of functional analysis, particularly in the study of the convergence of sequences of functions, and is often used in the context of nonlinear partial differential equations. The lemma provides conditions under which compactness can be guaranteed for a sequence of functions in certain function spaces. More specifically, it deals with the convergence properties of families of bounded sets in reflexive Banach spaces.