An optimal compactness theorem and application to elliptic-parabolic systems
The authors prove and apply the following theorem: Let \((V,\|\cdot \|)\), \((H,|\cdot |)\) be two separable Hilbert spaces. Assume that \(V\subset H\) with a compact and dense imbedding. Consider a sequence \(\{ u^\varepsilon\}_{\varepsilon >0}\) converging weakly to function \(u\) in \(L^2 (0, T;V)\), \(T<+ \infty\). Then \(u^\varepsilon\) converges strongly to \(u\) in \(L^2 (0, T;H)\) if and only ifNEWLINENEWLINENEWLINE1) \(u^\varepsilon (t)\) converges to \(u(t)\) weakly in \(H\) for a.e. \(t\),NEWLINENEWLINENEWLINE2) \(\lim_{\text{meas}(E) \to 0,E<\langle 0,T\rangle} \sup_{\varepsilon> 0}\int_E\bigl |u^\varepsilon (t)\bigr |^2dt=0.\)
scientific article; zbMATH DE number 1631135