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.
Best Answer
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.