Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain.
I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification sequence $u_\epsilon \in C^\infty(\mathbb{R};H^1)$ such that $u_\epsilon \to u$ in $L^2(0,T;H^1)$ and $u_\epsilon' \to u'$ in $L^2(0,T;H^{-1})$.
This sequence is smooth only in time. Is it possible to get a sequence from mollification that is also smooth in space?
Maybe I can approximate each $u_\epsilon(t)$ for each $t$ by spacial mollification? References would be appreciated.