Let $H$ be a Hilbert space such that $H\hookrightarrow \hookrightarrow L^2(\mathbb{R}^d)$. Let $\{u_n\}$ be a sequence such that $u_n$ converges strongly to $v$ in $C([0,T]; H')$ and converges weakly to u in $L^2((0,T) \times \mathbb{R}^d)$, converges in weak star sense to $u$ in $L^\infty((0,T) ; L^2(\mathbb{R}^d))$. Can we say that $u_n(t)$ converges weakly to $u(t)$ in $L^2(\mathbb{R}^d)$ and do we have $u \equiv v$?
We can observe that as evaluation map is continuous, we have strong convergence of $u_n(t) \to v(t)$ in $H'$.