This question is a follow up to this question.

Let $\Omega \subset \mathbb{R}^d$ be an open connected set. For each $t\in \mathbb{R}^+$ let $u_d:\partial\Omega \to \mathbb{R}$ be in $H^{1/2}(\partial \Omega)$. Let $u_e : \Omega \to \mathbb{R}$ be an extension of $u_d$ from the boundary to the interior.

I am interested in the set of all $u_d$ for which we can find an extension $u_e$ such that

$$ \left\| u_e\right\|_{L^2(\Omega)} + \left\| \nabla u_e\right\|_{L^2(\Omega)} + \left\|\frac{\mathrm{d}}{\mathrm{d}t} u_e\right\|_{L^2(\Omega)} $$

is finite for all $t\in \mathbb{R}^+$. What restrictions do I have to place on $u_d$ to have a well-defined time derivative of the extension? In addition, I am looking for a reference and a name for this space.