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

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  • $\begingroup$ 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 $L^2(\Omega)$. $\endgroup$ – gerw Mar 18 '14 at 19:29
  • $\begingroup$ @gerw But we already have $u_n \rightharpoonup u$ in $L^2(0,T;H^1)$. $\endgroup$ – maximumtag Mar 18 '14 at 19:31
  • $\begingroup$ Ok, so you should rewrite your last paragraph as "if we additionally have..." $\endgroup$ – gerw Mar 18 '14 at 19:32
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By using Aubin-Lions, you get $u_n \to u$ in $L^2(0,T; L^2(\Omega))$.

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  • $\begingroup$ Thanks. I don't know why I assumed we needed $u_n'$ in a stronger space. $\endgroup$ – maximumtag Mar 18 '14 at 19:33

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