# Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as,

\begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf B &= 0, &\text{No-name law (no monopoles),}\\\ \nabla \times \mathbf E &= - \partial_t \mathbf B, &\text{Faraday's law,}\\\ \nabla \times \mathbf B &= \mu \varepsilon\partial_t \mathbf E + \mu \mathbf J_f, &\text{Ampere's law}. \end{align} There are many equivalent formulations, for instance in terms of potentials and Gauges. My question is related to the regularity of the solution pair $(\mathbf E, \mathbf B)$. As the equations are hyperbolic and my knowledge is largely in elliptic equations (which seem to be completely different beasts to handle... I have heard: "Partial differential equations are like a zoo, even if the animals look the same you might have to treat them differently").

Regularity questions:

1. What are the standard references for the regularity (of the solutions) of the Maxwell equations?
2. If we have the equations on domains, what is the dependence of the regularity of the solutions in terms of the regularity of the boundary?
3. Which formulations are most convenient to prove regularity properties for hyperbolic equations? As I have said above, there exist many equivalent ones.
4. Is there any work done, and what work, on the regularity questions for the Maxwell equations in a functional analytic framework? Here I mean phrasing the equations as a ordinary differential equations in a Banach space, just as we would have the analysis of the heat kernel as a convolution-type operator for the heat equation. How about harmonic analysis?
5. Has there been any work done on the Maxwell equations in terms of gradient flows on metric spaces (as in the work of Felix Otto et al., for the Fokker-Planck equation, sorry, the Ornstein-Uhlenbeck process)?

Before the question gets closed before it is "overly broad, rhetoric or whatever", please note that my question is mainly about the regularity for Maxwell equations and if one of the other questions can get answered or get pointed to a reference in the process, that would be nice. My background in PDE is mainly from the elliptic side, I do not have much knowledge about their hyperbolic ones, other than the trivial results.

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There are a few, not many, books on hyperbolic equations. You might have a look to that of S. Benzoni-Gavage and myself: Multi-dimensional hyperbolic partial differential equations. First order systems and applications, Oxford Mathematical Monographs, Oxford University Press (2007).

A basic fact of hyperbolic systems of PDEs is that the Cauchy problem is well-posed in both directions of time. Therefore the regularity of the solution cannot be improved as time increase, contrary to the parabolic case. Also, this implies that such systems cannot be recast as gradient flows; instead, some of them can be reformulated as Hamiltonian system (say, if the semi-group is reversible).

That said, there exists nevertheless some regularity properties. On the one hand, the singularities are polarized. This means that the solution is smooth along non-characteristic directions, and most of (but not all) the solution is smooth even in characteristic directions. Let me take the example of the wave equation $$\partial_t^2u=\Delta_x u$$ in ${\mathbb R}^{1+d}$. Then the wave-front set is invariant under the bi-characteristic flow $$\frac{dx}{dt}=p,\qquad\frac{dp}{dt}=-x.$$ A by-product (which can be proved directly by an integral formula of the solution) is that if the initial data $u(t=0,\cdot)$, $\partial_tu(t=0,\cdot)$ is smooth away from $x=0$, then the solution is smooth away from $|x|=|t|$. However the wave-frontset approach tells you much more.

On another hand, the decay of the initial data at infinity implies some space-time integrability of the solution. These properties are not directly related to hyperbolicity. They are consequences of the dispersion. In the case of the wave equation, this is the fact that the characteristic cone $|x|=|t|$ has not flat part. Such integrability statements are know as Strichartz-like inequalities.

Finally, the ODE point of view is adopted by Klainerman, Machedon, Christodoulou and others, mixed with Strichartz inequalities, to prove the well-posedness of the Cauchy problem for semi-linear hyperbolic systems, like Einstein equations of general relativity.

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So, the statement you make is that the $C_0$-semigroup would actually be a group? As in the Schrödinger equation, which is a Weyl rotation of the heat equation? I understood the 'cannot be made smoother' in this context as the fact that you can 'go back in time' and would make things less smooth. I guess this is what you are saying but more mathematically. I still have to wrap my head around why a rotation in the complex plane would give such profound consequences in the behavior of the heat equation. Perhaps, that is for another question when I thought more about it. (...) –  Jonas Teuwen Nov 11 '12 at 17:18
(...) Klainerman is the name I have heard before in this context. I will look them up (I have seen the Strichartz-like estimates before). Thanks for the reference, I will request your book from the library. Nice to have a real expert answering your question online. –  Jonas Teuwen Nov 11 '12 at 17:19
@Jonas. You interpret correctly. –  Denis Serre Nov 11 '12 at 19:46