Does $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ give something useful?

Question

If $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$ and $u_n' \rightharpoonup u'$ in $L^2(0,T;H^{-1})$ is there any way to extract a strongly convergent subsequence of $u_n$ in some useful space for PDEs? Or some a.e. convergence result?

For example, if we had the time derivatives weakly convergent in the space $L^2(0,T;L^2)$ then we can use Lions-Aubin to get $u_n \to u$ in $L^2(0,T;L^2)$.

Answer 1

By using Aubin-Lions, you get $u_n \to u$ in $L^2(0,T; L^2(\Omega))$.