Let $X$ be a bounded domain.
We have a sequence $b_n(t) \to b(t)$ pointwise a.e. (no dependence on space), and functions $f_n \to f$ in $L^2(0,T;L^2(X))$ such that $b_n\nabla f_n \rightharpoonup g$ in $L^2(0,T;L^2(X))$.
We know that $\nabla f \in L^2(0,T;L^2(X))$, and also each $b_n \in L^\infty(0,T)$ (but the bound is not uniform).
Is it possible to say that $g=b\nabla f$?
Note that $b_nf_n \to bf$ pointwise a.e. But we don't know if $bf \in L^2(0,T;L^2)$ or not. And we don't know anything about boundedness of $b$ a priori.