# 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?

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 $L^2(0,T;L^2)$.

• Your last paragraph is wrong. In order to apply Lions-Aubin, you need additionally the weak convergence of the function itself in $L^2(0,T;X)$, where $X$ is compactly embedded in $L^2(\Omega)$. – gerw Mar 18 '14 at 19:29
• @gerw But we already have $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$. – maximumtag Mar 18 '14 at 19:31
• Ok, so you should rewrite your last paragraph as "if we additionally have..." – gerw Mar 18 '14 at 19:32

By using Aubin-Lions, you get $u_n \to u$ in $L^2(0,T; L^2(\Omega))$.
• Thanks. I don't know why I assumed we needed $u_n'$ in a stronger space. – maximumtag Mar 18 '14 at 19:33