Energy methods for first order systems

Question

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 !

Answer 1

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 ${\rm div}H={\rm div}E=0$.

Answer 2

The literature on first order hyperbolic PDEs is vast.

Some places where you can get some answers:

• Courant and Hilbert, Methods of mathematical physics, Vol II.
• K.O. Friedrichs, "Symmetric hyperbolic linear differential equations", CPAM 1954
• T. Kato, "Linear evolution equations of hyperbolic type", J. Fac. Sci. Univ. Tokyo 1970; "Linear evolution equations of hyperbolic type II", J. Math. Soc. Japan 1973; "The Cauchy problem for quasi-linear symmetric hyperbolic systems" (Arch. Ration. Mech. Anal.)