Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$.
It follows that for almost all $t$, $u_n(t)$ is bounded in $L^\infty(\Omega)$, so we can extract a weak-* convergent subsequence and after a bit of work we can show that $$\text{for almost all $t$} \qquad u_n(t) \rightharpoonup u(t) \text{ in $L^q(\Omega)$} \qquad\text{for all $q < \infty$.}$$
I want to show that this weak convergence holds for all $t$.
From personal correspondence, I have been told that
The key point is that one already has convergence in the larger space. So one already knows that pointwise, i.e. for all $t$, $u(t)$ converges in this larger space. But since it is also bounded in $L^\infty$, $u(t)$ [for almost all $t$ -- riem's note] will also converge weak-* up to a subsequence, and the limit is the same are identified as distributions. The same is true for any $L^q$, $q<\infty$.
Simply put, I don't get why it must hold for all $t$. I think we need to exploit the continuity of $u_n$ and $u$ wrt. $t$ in the larger space $H^{-1}$ but once again $L^q$ is a stronger space than this..
