I am currently working on hyperbolic equations on bounded domains. For this reason, I am considering in particular functions $u$ such that:
$u \in L^2(0;T;H^1_0(\Omega)), \quad u' \in L^2(0;T;L^2(\Omega)), \quad u'' \in L^2(0;T;{H^1_0(\Omega)}')$,
where $\Omega$ is the bounded domain I am working on and ${H^1_0(\Omega)}'$ is the dual of $H^1_0(\Omega)$.
Such functions satisfy in particular:
$u \in \mathcal{C}^0([0;T];L^2(\Omega)), \quad u' \in \mathcal{C}^0([0;T];{H^1_0(\Omega)}')$.
That is not very complicated to see. However, apparently we even have the following better result:
$u \in \mathcal{C}^0([0;T];H^1_0(\Omega)), \quad u' \in \mathcal{C}^0([0;T];L^2(\Omega))$.
I have absolutely no idea how to prove this. It seems to be a general result on Gelfand triples but I haven't found any satisfying ressource explaining this and keeping it to a reasonable level up to now .
If anyone has an idea, don't hesitate to share. Thank you very much. Boris.