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:**

- What are the standard references for the regularity (of the solutions) of the Maxwell equations?
- 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?
- Which formulations are most convenient to prove regularity properties for hyperbolic equations? As I have said above, there exist many equivalent ones.
- 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?
- 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.