Suppose we have proved the equation $$u_{tt}-u_{xx}=0\quad\text{in}\quad(0,T)\times(0,\ell)$$ with some boundary and initial conditions has an unique solution $u\in C^2(0,T,H^2(0,\ell))$ and we need a $L^2$-estimate to $u_{tt}$.
Differentiating the equation with respect to $t$, multiplying by $u_{tt}$ and integrating with respect to $x$ we get
\begin{align*} \int_0^\ell u_{ttt}u_{tt}\;dx-\int_0^\ell u_{txx}u_{tt}\;dx&=0\\ \int_0^\ell u_{ttt}u_{tt}\;dx+\int_0^\ell u_{tx}u_{ttx}\;dx&=0\\ \frac{1}{2}\frac{d}{dt}\int_0^\ell \Big((u_{tt})^2 +(u_{tx})^2\Big)\;dx&=0\\ \|u_{tt}\|_{L^2}^2\leq \int_0^\ell \Big((u_{tt})^2 +(u_{tx})^2\Big)\;dx&=C \text{ (constant)} \end{align*}
Notice that only second order terms appear in the final estimate, so it makes sense. However, the calculation seems to make no sense because we have used third order terms that are not defined.
How can we justify calculations like that? I've seen the following comment: the differentiation with respect to $t$ is formally not allowed, but smoothing the initial data and going to the limit in the final estimates justifies the calculation. However I need some more details and references.
Thanks.
I have an answer to this but it is sort of particular to one way of thinking about PDEs. I think other methods yield similar results, but I am not as familiar with how they justify them. So!
To be specific, semigroup theory says something like if you want to solve $U_t = A u$ where $A$ is an unbounded operator on a Hilbert space $H$ with domain $D(A)$ (satisfying necessary assumptions - see the Hille-Yosida or Lumer-Phillips theorems) and initial conditions $U(0) = U_0$, you can get greater degrees of time differentiability of the solution $U(t)$ by asking for more "smoothness" (membership in the domain of higher powers of $A$) from the initial data. Specifically, something like $$U(t) \in C^{k-j}\left(0,T;D(A^j) \right)$$ Provided that $U_0 \in D(A^k)$. Then you formally can compute stability results through multipliers to the original equation, even though the highest time derivative is probably not defined for data that is just in $H$, i.e. $D(A^0)$. You can then justify that the results hold for initial data that is merely in $H$ by a limit process, taking an approximating sequence of smooth initial data to the desired nonsmooth initial data.
This is of course a very vague overview of a huge topic! I'm not sure what your background is, so maybe this is all old news to you and you were asking something else. If you would like some reading, the classic text is "Semigroups of linear operators and applications to partial differential equations" by Pazy, but it is a bit dry. I quite liked Functional Analysis, Sobolev Spaces and Partial Differential Equations by Brezis. There are many other sources as well!