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We study the Cauchy problem for the $3D$ compressible Euler equations underan arbitrary equation of state with positive speed of sound, aside from that ofa Chaplygin gas. For open sets of smooth initial data with non-trivialvorticity and entropy, our main results yield a constructive proof of theformation, structure, and stability of the singular boundary, which is the setof points where the solution forms a shock singularity, i.e., where somefirst-order Cartesian coordinate partial derivatives of the velocity anddensity blow up. We prove that in the solution regime under study, the singularboundary has the structure of a degenerate, acoustically null $3D$submanifold-with-boundary. Our approach yields the full structure of aneighborhood of a connected component of the crease, which is a $2D$acoustically spacelike submanifold equal to the past boundary of the singularboundary. In the study of shocks, the crease plays the role of the "trueinitial singularity" from which the singular boundary emerges, and it is acrucial ingredient for setting up the shock development problem. These are thefirst results revealing the totality of these structures without symmetry,irrotationality, or isentropicity assumptions. Moreover, even within thesub-class of irrotational and isentropic solutions, these are the firstconstructive results revealing these structures without a strict convexityassumption on the shape of the singular boundary. Our proof relies on a newmethod: the construction of rough foliations of spacetime, dynamically adaptedto the exact shape of the singular boundary and crease, where the latter isprovably two degrees less differentiable than the fluid.

The emergence of the singular boundary from the crease in $3D$ compressible Euler flow