In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system $$u_t+\sum_{j=1}^nB_ju_{x_j}=f\quad\mbox{in}\quad\Bbb R^n\times(0,T),$$ $$u|_{t=0}=g,$$ where $u:\Bbb R^n\to\Bbb R^m$, and (for the purpose of this question) $f,g$ are sufficiently smooth (with decay at infinity) and the $B_j$ are constant.
To achieve his goal he considers the following parabolic problem for $\varepsilon>0$ prove existence of a weak solution to the following linear parabolic system $$u^\varepsilon_t+\sum_{j=1}^nB_ju^\varepsilon_{x_j}-\varepsilon\Delta u^\varepsilon=f\quad\mbox{in}\quad\Bbb R^n\times(0,T),$$ $$u|_{t=0}=g.$$ Standard parabolic theory yields a unique weak solution $u^\varepsilon\in L^2(0,T;H^3(\Bbb R^n;\Bbb R^m))$, with $u_t\in L^2(0,T;H^1(\Bbb R^n;\Bbb R^m))$. Moreover, we have that $\max_{0\le t\le T}\|u^\varepsilon(t)\|_{L^2(\Bbb R^n;\Bbb R^m)}$ is uniformly bounded independent of $\varepsilon$.
He now proves a uniform bound independent of $\varepsilon$ for $u^{\varepsilon}_t$, but to do so he differentiates the above equation with respect to $t$, letting $v=u^\varepsilon_t$, obtaining $$v_t+\sum_{j=1}^nB_jv_{x_j}-\varepsilon\Delta v=f_t\quad\mbox{in}\quad\Bbb R^n\times(0,T),$$ $$v|_{t=0}=f(0)+\varepsilon \Delta g-\sum_{j=1}^mB_jg_{x_j}.$$ By the current theory, the above equation yields a unique solution $w$ which satisfies the desired estimate for $\max_{0\le t\le T}\|w(t)\|_{L^2(\Bbb R^n;\Bbb R^m)}$, independent of $\varepsilon$.
My question is: (a) what allows him to differentiate the equation?
(b) if he can differentiate the equation, why is it necessarily true that $w=u^\varepsilon_t$?