Lets assume we have $b(x,t) \in L^\infty$ periodic in x and $\frac{db}{dt} \in L^{p}, p\in\mathbb{N}$.
$b(x,t)$ converges weakly to an $f(t)=\int b(x,t)dx \in L^q$ for all $1<q<\infty$.
Also, $\frac{db}{dt}$ converges weakly-* in $L^{p}$.
Does this mean that $\frac{db}{dt}$ is converging weakly-* to $\frac{df}{dt}$? Or can one achieve this with another assumption?