I will discuss recent work, joint with Leo Abbrescia and Dongxiao Yu, on the 3D isentropic compressible Euler equations in spherical symmetry. Under any equation of state except the Chaplygin gas, we study open sets of smooth “asymptotically flat” data on R3 that are perturbations of a non-vacuum constant state. Our main result is that these data launch a unique Maximal Globally Hyperbolic Development (MGHD), where singularities develop on part of the boundary. These are the first stable, global-in-space blowup + uniqueness results of their type for any quasilinear hyperbolic PDE on R3 with asymptotically flat data. Roughly speaking, an MGHD is a largest possible classical solution that is uniquely determined by the data. Explicit examples for related quasilinear hyperbolic PDEs show that MGHD-uniqueness cannot be determined locally in spacetime, and can in fact fail for some equations/some data. For these reasons, our proof of MGHD-uniqueness relies on the fact that we can construct it in its entirety and derive the complete structure of its boundary, all the way out to spatial infinity, and all the way down to the center of symmetry. In particular, the boundary of our MGHDs consists of smooth curves continuously joined at a corner: a singular boundary that extends out to spatial infinity, along which the fluid’s gradient blows up, and a Cauchy horizon that extends all the way to the center of symmetry, along which the solution remains smooth. The techniques we use are robust and are applicable to a wide class of quasilinear hyperbolic PDEs that fail to satisfy the null condition.