An alternative proof of the Aubin-Lions lemma
\noindent The author uses a generalized version of the Weyl-Riesz compactness criterion of subsets of Lebesgue-Bochner spaces (due to Gutman) to give a new proof of the Aubin-Lions theorem. The Aubin-Lions theorem is: Let \(X_0, X_1, X\) be Banach spaces satisfying \(X_0 \subset X \subset X_1\) with continuous embeddings. Let \(T>0\) be finite, and let \(\alpha_0, \alpha_1 \in [1, \infty)\). Consider the space \[ \mathcal{Y} := \{ v \in L^{\alpha_0}(0, T; X_0): v^{\prime} \in L^{\alpha_1}(0, T; X_1) \} \] which is a Banach space with the obvious norm. If the embedding \(X_0 \hookrightarrow X_1\) is compact, and if \(\alpha_0\), \(\alpha_1\) are such that if \(\alpha_1 = 1\), then \(\alpha_0 = 1\), the embedding \(\mathcal{Y} \hookrightarrow L^{\alpha_0}(0, T; X)\) is also compact. The Aubin-Lions theorem is used in proofs of the existence of solutions of nonlinear evolution equations.
scientific article; zbMATH DE number 6226073
scientific article