  These lecture notes accompany two classes given at the NRHEP2 school. In the
first lecture I introduce the basic concepts used for analyzing well-posedness,
that is the existence of a unique solution depending continuously on given
data, of evolution partial differential equations. I show how strong
hyperbolicity guarantees well-posedness of the initial value problem. Symmetric
hyperbolic systems are shown to render the initial boundary value problem
well-posed with maximally dissipative boundary conditions. I discuss the
Laplace-Fourier method for analyzing the initial boundary value problem.
Finally I state how these notions extend to systems that are first order in
time and second order in space. In the second lecture I discuss the effect that
the gauge freedom of electromagnetism has on the PDE status of the initial
value problem. I focus on gauge choices, strong-hyperbolicity and the
construction of constraint preserving boundary conditions. I show that strongly
hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I
examine which of these formulations is additionally symmetric hyperbolic and
finally demonstrate that the system can be made boundary stable.
