Clarifications about a weakly convergent seaqence

Question

I came across a situation in a paper page 36 where I need some clarifications. I have a weakly convergent sequence $(a_n)$ in the space $L^2(0,T;L^2(\Omega))$, which converges weakly to $a$. The paper proves that $(b_n)$ is a bounded sequence in the same space, then concludes that: $$ \int_0^T \langle a_n(t)-a,b_n(t) \rangle_{L^2(\Omega)} \mathrm dt \longrightarrow 0 $$ I'm trying to understand the reasoning behind this result. Could someone please explain why this integral tends to zero as $n$ approaches infinity?

Answer 1

There is no reason behind this, as the claim as written in your question is wrong. (To see that this is not true, take $a_n=b_n$ weakly but not strongly converging to zero.)

In addition, nothing in the paper suggests that the authors are using some unmentioned compactness. In my opinion, the proof is wrong and cannot be corrected without adding assumptions on $N_j$. One remedy would be to require $N_j \in \mathcal L(Y,Y)$ (instead of $N_j \in \mathcal L(V,Y)$). At least the proof could be salvaged, as then $Ny_k \to N \bar y$ strongly in $L^2(0,T;Y)$ by Aubin-Lions lemma. But then the examples in Section 8 fail to satisfy this requirement.