weak convergence and compactness

Question

please how can I prove that if a sequence $u_n \to u$ in $L^{\infty}(\mathbb{R}^+; H^1(\Omega)) $ weak * and $\partial_t u_n \to \partial_t u$ in $L^2(0,T, L^2(\Omega)) $ weak for all $T>0$ implies that $u_n \to u$ in $L^2(0,T,L^2(\Omega))$ strong for all $T>0$.

Answer 1

Since weakly convergent sequences are norm bounded in the respective space, we have in particular

$$\|u_n\|_{L^2(0,T;H^1)}, \|\partial_t u_n\|_{L^2 (H^1)'} \leq C$$

hence by the Aubin--Lions lemma you get that $u_n \to u$ in $L^2(0,T;L^2)$ strongly. For the proof I refer you to the paper by Jacques Simon. Alternatively, you can look at the book from Boyer and Fabrie.