Timeline for $u_n$ bounded in $L^\infty(0,T;H) \cap L^2(0,T;V)$ implies $u_n \to u$ strongly in $L^2(0,T;H)$?
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9 events
| when toggle format | what | by | license | comment | |
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| Mar 4, 2015 at 10:08 | comment | added | Charpe | Ok I think I understand. The bound gives $u_{n_{j}(t)}(t) \rightharpoonup u(t)$ in $L^2$ for a.e. $t$, but the subsequence depends on $t$. So the Theorem cannot be applied for even a subsequence, and my desired result does not hold even for a subsequence (a subsequence convergence is enough for PDE type arguments). It is strange that this theorem is used in papers without considering such issues. | |
| Mar 4, 2015 at 9:20 | comment | added | Jean Van Schaftingen | @Charpe The uniform bound in $L^\infty (0, T; L^2)$ only implies that for almost every $t \in [0, T]$, a subsequence converges weakly; since the set $[0, T]$ is not contable, you cannot conclude by a diagonal argument. | |
| Mar 4, 2015 at 8:06 | vote | accept | Charpe | ||
| Mar 4, 2015 at 8:05 | comment | added | Charpe | Oh I see. Well, we certainly have the equiintegrability by Remark 1, point 2. I had thought that the uniform bound on $u_n$ in $L^\infty(0,T;L^2)$ gave $u_n(t) \rightharpoonup u(t)$ in $H=L^2$ a.e. $t$. | |
| Mar 4, 2015 at 7:25 | history | edited | Jean Van Schaftingen | CC BY-SA 3.0 |
Explained L^\infty convergence
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| Mar 4, 2015 at 7:19 | comment | added | Jean Van Schaftingen | The theorem that you mention is quite different: it assumes that for almost every $t \in [0, T]$, the sequence $(u_n)$ converges weakly in $H$ and that it is equiintegrable in $L^2 (0, T; H)$. This is not a consequence of the weak-* $L^\infty (0, T; V)$ convergence. | |
| Mar 3, 2015 at 19:35 | comment | added | Charpe | I am of course aware of the classical example involving the sine that weak convergence does not imply strong convergence. But I don't see why your example converges weakly in the $L^\infty(0,T;L^2)$ either. | |
| Mar 3, 2015 at 19:34 | comment | added | Charpe | Thank you, but doesn't this contradict Theorem 1 of this paper? | |
| Mar 3, 2015 at 7:35 | history | answered | Jean Van Schaftingen | CC BY-SA 3.0 |