On regularity of Hyperbolic Equations In my PDE class we are following Evans PDE book, we were reading about Regularity of weak solutions for Hyperbolic equations, more specific in the proof theorem 5 section 7.2.3., the author states that we have
\begin{equation}
\frac{d}{dt}(\|\tilde{u}_m^{'}\|_{L^2(U)}^2+A[\tilde{u}_m,\tilde{u}_m])\leq C(\|\tilde{u}_m^{'}\|_{L^2(U)}^2+A[\tilde{u}_m,\tilde{u}_m]+\|f^{'}\|_{L^{2}(U)}^2)
\end{equation}
where $\tilde{u}_m=u_m^{'}$, also the estimate
\begin{equation}
\|u_m\|_{H^2(U)}^2\leq C(\|f\|_{L^2(U)}^2 +\|u_m^{''}\|_{L^2(U)}^2+\|u_m\|_{L^2(U)}^2)
\end{equation}
Evans says that using this last inequality in the first and aplying Gronwall's Inequality we deduce that
\begin{equation}
\sup_{0\leq t\leq T}(\|u_m(t)\|_{H^2(U)}^2+\|u_m^{'}(t)\|_{H^1(U)}^2+\|u_m^{''}(t)\|_{L^2(U)}^2)\leq C(\|f\|_{H^1(0,T;L^2(U))}^2+\|g\|_{H^2(U)}^2+\|h\|_{H^1(U)}^2)
\end{equation}
My problem is that I don't understand how this last expression is obtained, can anyone help me?
Edit:
We are looking about regularity of weak solutions of the PDE
\begin{equation}
\begin{array}[rcl]
 fu_{tt}+Lu&=f& \text{in } U_{T},\\
&u=0&\text{in } \partial U\times[0,T],\\
&u(0)=g&\text{in } U\times\{t=0\}\\
&u^{'}(0)=h&\text{in } U\times\{t=0\}\\
\end{array}
\end{equation}
we know that if $f\in L^2(0,T;L^(U))$, $g\in H_0^1(U)$ and $h\in L^2(U)$ there exist a weak solution of this PDE, for regularity we are asuming that $f,g$ and $h$ are in their spaces respectively and moreover $f^{'}\in L^2(0,T;L^2)$, $g\in H^2(U)$ and $h\in H_0^1(U)$. Hope this clarify about my question.
 A: I am dropping the subscript $m$ which is used to indicate approximating solutions.
The first inequality (with the time derivative on the left) comes from considering the pde that is satisfied by $\tilde u = u'$ and applying the usual energy estimate. Apply a Gronwall argument here to obtain an estimate
$$
\sup_t \left(\|\tilde u'(t)\|^2_{L^2} + A(\tilde u(t), \tilde u(t)) \right) \\  \quad \le C\left( \|\tilde u'(0)\|^2_{L^2} + A(\tilde u(0), \tilde u(0)) + \int_0^T \|f'\|^2_{L^2} \right)
$$
You read off from the pde for $\tilde u$ what $\tilde u(0)$ and $\tilde u'(0)$ must be. This implies estimates for
$$
\sup_t \left(\| u_{tt}(t)\|_{L^2} + \| u_t(t)\|_{H^1} \right)
$$
since the form $A$ is (essentially) coercive.
The second inequality follows from the pde itself plus elliptic regularity theory for the operator $L$. Just write $Lu = -u_{tt} + f$ and use an estimate like
$$
\|u\|_{H^2} \le C(\|Lu\|_{L^2} + \|u\|_{L^2})
$$
which surely appears in an earlier chapter of the book.
Since you already have an estimate for $\|u_{tt}\|_{L^2}$, the desired estimate now can be derived. Just keep track of where norms of $g$ and $h$ enter the estimates.
