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

Abstract

We study the Cauchy problem for the $3D$ compressible Euler equations under an arbitrary equation of state with positive speed of sound, aside from that of a Chaplygin gas. For open sets of smooth initial data with non-trivial vorticity and entropy, our main results yield a constructive proof of the formation, structure, and stability of the singular boundary, which is the set of points where the solution forms a shock singularity, i.e., where some first-order Cartesian coordinate partial derivatives of the velocity and density blow up. We prove that in the solution regime under study, the singular boundary has the structure of a degenerate, acoustically null $3D$ submanifold-with-boundary. Our approach yields the full structure of a neighborhood of a connected component of the crease, which is a $2D$ acoustically spacelike submanifold equal to the past boundary of the singular boundary. In the study of shocks, the crease plays the role of the "true initial singularity" from which the singular boundary emerges, and it is a crucial ingredient for setting up the shock development problem. These are the first results revealing the totality of these structures without symmetry, irrotationality, or isentropicity assumptions. Moreover, even within the sub-class of irrotational and isentropic solutions, these are the first constructive results revealing these structures without a strict convexity assumption on the shape of the singular boundary. Our proof relies on a new method: the construction of rough foliations of spacetime, dynamically adapted to the exact shape of the singular boundary and crease, where the latter is provably two degrees less differentiable than the fluid.