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 !



Energy methods for first order systems
Sorry for self-advertising. These system are called symmetric (in the sense of Friedrich) hyperbolic. More generally, a first-order symmetric hyperbolic system has the form
$$A^0\partial\_tu+\sum\_{\alpha=1}^dA^\alpha\partial\_\alpha u=0,$$
where $A^\alpha$ are symmetric $n\times n$ real matrices and $A^0$ is symmetric positive definite. If the physical domain is the whole space ${\mathbb R}^d$, then the Cauchy problem is well-posed in $L^2({\mathbb R}^d)^n$, and more generally in $H^s({\mathbb R}^d)^n$ for every $s\in{\mathbb R}$. The energy 
$$\int\_{{\mathbb R}^d}u^TA^0u\,dx$$
is independent of the time variable. By the way, in your examples, $A^0$ is the identity matrix. 


An appropriate reference is the first chapter of our book



> 
> S. Benzoni-Gavage & D. Serre. *Multi-dimensional hyperbolic partial differential equations. First order systems and applications.*
>  Oxford Mathematical Monographs, Oxford University Press (2007, ISBN-10: 0-19-921123-X, ISBN 13: 978-0-19-921123-4).
> 
> 
> 


In hyperbolic systems, waves travel at finite speeds. If $\xi\in S^{d-1}$, the propagation speeds $\lambda\_1(\xi),\ldots,\lambda\_n(\xi)$ are the (generalized) eigenvalues:
$$\det(A(\xi)-\lambda\_j(\xi)A^0)=0,\qquad A(\xi):=\sum\_\alpha\xi\_\alpha A^\alpha.$$
If $r\_j(\xi)$ is a corresponding eigenvector, then $u(t,x)=\phi(x-\lambda\_j(\xi)t)r\_j$ is an exact solution of the system, for every function $\phi$ of a real variable. 


The example of the Maxwell's equations is especially interesting. You have $A^0=I\_6$ and 
$$A(\xi)=\begin{pmatrix} 0\_3 & J(\xi) \\ -J(\xi) & 0\_3 \end{pmatrix},\qquad J(\xi):=\begin{pmatrix} 0 & \xi\_3 & -\xi\_2 \\ -\xi\_3 & 0 & \xi\_1 \\ \xi\_2 & -\xi\_1 & 0 \end{pmatrix}.$$
One therefore finds the velocities $\pm1,0$ in every direction $\xi$ (notice that you normalized $c=1$). The zero velocities looks strange and is actually spurious. It is ruled out by the constraint ${\rm div}H={\rm div}E=0$.


