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 !