  In this article, we provide notes that complement the lectures on the
relativistic Euler equations and shocks that were given by the second author at
the program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime,
which was hosted by the Erwin Schrodinger International Institute for
Mathematics and Physics in Vienna in February, 2022. We set the stage by
introducing a standard first-order formulation of the relativistic Euler
equations and providing a brief overview of local well-posedness in Sobolev
spaces. Then, using Riemann invariants, we provide the first detailed
construction of a localized subset of the maximal globally hyperbolic
developments of an open set of initially smooth, shock-forming isentropic
solutions in 1D, with a focus on describing the singular boundary and the
Cauchy horizon that emerges from the singularity. Next, we provide an overview
of the new second-order formulation of the 3D relativistic Euler equations
derived in [41], its rich geometric and analytic structures, their implications
for the mathematical theory of shock waves, and their connection to the setup
we use in our 1D analysis of shocks. We then highlight some key prior results
on the study of shock formation and related problems. Furthermore, we provide
an overview of how the formulation of the flow derived in [41] can be used to
study shock formation in multiple spatial dimensions. Finally, we discuss
various open problems tied to shocks.
