I have trouble to find references on the use of energy method (Faedo-Galerkin method using approximation in finite dimension space) to solve problems of the form $$ \begin{cases} \partial_t u = \mathcal B v \\ \partial_t v = -\mathcal B^* u \end{cases}$$ where $\mathcal B$ and $\mathcal B^*$ are operators that are (at least formally) adjoint.
The two examples I have in mind is the wave equation $$ \begin{cases} \partial_t p = \mathrm{div}\, \mathbf u \\ \partial_t \mathbf u = \mathrm{grad}\, p \end{cases}$$ or the Maxwell equation $$ \begin{cases} \partial_t \mathbf E = \mathrm{rot}\, \mathbf H \\ \partial_t \mathbf H = - \mathrm{rot}\, \mathbf E \end{cases}$$
I can't find any reference on the well-posedness of such systems using energy method. I can find a lot about first order parabolic evolution equations or second order evolution equations (for instance in the classical books Mathematical Analysis and Numerical Methods for Science and Technology of Dautray and Lions or even Evan's book) but nothing on first order systems for these wave-like problems.
Can you give me some references or telling me why it is not possible if I miss something ?
Thanks !