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In this article, we provide notes that complement the lectures on therelativistic Euler equations and shocks that were given by the second author atthe program Mathematical Perspectives of Gravitation Beyond the Vacuum Regime,which was hosted by the Erwin Schrodinger International Institute forMathematics and Physics in Vienna in February, 2022. We set the stage byintroducing a standard first-order formulation of the relativistic Eulerequations and providing a brief overview of local well-posedness in Sobolevspaces. Then, using Riemann invariants, we provide the first detailedconstruction of a localized subset of the maximal globally hyperbolicdevelopments of an open set of initially smooth, shock-forming isentropicsolutions in 1D, with a focus on describing the singular boundary and theCauchy horizon that emerges from the singularity. Next, we provide an overviewof the new second-order formulation of the 3D relativistic Euler equationsderived in [41], its rich geometric and analytic structures, their implicationsfor the mathematical theory of shock waves, and their connection to the setupwe use in our 1D analysis of shocks. We then highlight some key prior resultson the study of shock formation and related problems. Furthermore, we providean overview of how the formulation of the flow derived in [41] can be used tostudy shock formation in multiple spatial dimensions. Finally, we discussvarious open problems tied to shocks.

The relativistic Euler equations: ESI notes on their geo-analytic structures and implications for shocks in $1D$ and multi-dimensions