Going from a weak formulation to a pointwise a.e statement; don't understand text (PDEs, sobolev spaces) I just read this:

For $u \in H^1(Q)$ where $Q=\cup_{t \in (0,T)}\Omega \times \{t\}$, we have that
  $$\int_{\Omega}u_tv +\nabla u \cdot \nabla v = \int_{\Omega}fv\quad\text{for all $v \in H^1(\Omega).$}\tag{1}$$

Now I want a bound on $u_t$ in $L^2(0,T;L^2(\Omega))$. The issue is we cannot take $v=u_t$ in (1) because $u_t \notin L^2(0,T;H^1(\Omega))$, it's only in $L^2(0,T;L^2(\Omega))$. So the author gets around this by the following argument.

From (1) it follows that $$u_t -\Delta u = f\quad\text{holds a.e. in $Q$.}$$  Now we can multiply this equation by $u_t$ and integrate over space, ....

This makes no sense to me. I know that the author has integrated by parts but $u$ is not smooth enough to have $-\Delta u$ make sense in a pointwise fashion; it only exists as a functional. So I don't understand why this a correct approach. 
The source is Variational methods in the stefan problem by José-Francisco Rodrigues, in the proof of Proposition 4.7.
Note that $u \in H^1(Q)$ if $u \in L^2(0,T;H^1(\Omega))$ and $u_t \in L^2(0,T;L^2(\Omega))$.
 A: So $\Delta u$ is still only defined in a weak sense, in that there is a distribution $w$ such that
$$
\int u \Delta \phi = \int w \phi
$$
for all $\phi \in C_0^\infty$.
So, if 
$$
\int_\Omega u_t \phi + \nabla u \cdot \nabla \phi = \int_\Omega f\phi
$$
for all $\phi$, then integrating by parts
$$
\int_\Omega u_t \phi - u \Delta \phi = \int_\Omega f\phi
$$
(there was a boundary term that vanished because $\phi$ is compactly supported), and then using the definition of the weak derivative
$$
\int_\Omega u_t \phi - w \phi = \int_\Omega(u_t - w)\phi = \int_\Omega f\phi
$$
and so, as distributions, you must have $u_t - w = f$.
Then, if $f \in L^2$ (and if $u_t - w$ is locally integrable?), the two sides are equal as distributions if and only if they are equal to each other (as functions) almost everywhere. (stated here, with a reference to Hormander)
A: Rearrange the weak formulation and you have
$$\int_\Omega\nabla u\cdot\nabla v=\int_\Omega (f-u_t)v,$$
with the necessary assumption that $f\in L^2(0,T;L^2(\Omega))$, $u(t)$ can be considered to be the weak solution of Laplace's equation with right had side in $L^2(\Omega)$, for  a.e. $t\in(0,T)$. Elliptic regularity indeed justifies that either $u(t)\in H^2(\Omega)$ (if the boundary of $\Omega$ is at least $C^2$), or if $\partial\Omega$ is not as nice (say Lipschitz), then $u(t)\in H^2_{loc}(\Omega)$. 
Either is sufficient to justify integrating by parts against a $C^\infty_c(Q)$ test function, which gives the equality a.e.
