Does anyone know this space? The author plays in an the article, it seems to me to be Banach and reflexive. Knowing his name already helps a lot, because he uses the norm of this space in the accounts.
$\Omega \subset \mathbb{R}^{n}$ is an open bounded set.
$I \subset \mathbb{R}$ is an open bounded interval.
$$\color{blue}{\boxed{V(\Omega \times I):=L^{\infty}(I;L^{2}(\Omega))\cap L^{p}(I;W^{1,p}(\Omega))}}$$