This work is joint with Leo Abbrescia and Dongxiao Yu. I will discuss two theorems for the 3D irrotational and isentropic compressible Euler equations in spherical symmetry. In the first theorem, for any equation of state besides that of the Chaplygin gas, we study the Cauchy problem for a set of smooth, small, “asymptotically flat” data on $\mathbb{R}^3$ that are perturbations of a non-vacuum constant state. We prove that these data launch a unique Maximal Globally Hyperbolic Development (MGHD), where gradient singularities develop on part of the boundary. This is the first stable, global-in-space blowup + uniqueness-result of its type for any quasilinear hyperbolic PDE on $\mathbb{R}^3$ 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 PDEs show that MGHD-uniqueness cannot be determined locally in spacetime, and in fact can fail for some equations/some data. Therefore, 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. 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.

In the second theorem, we study the Cauchy problem for a different, closely related set of smooth, small, asymptotically flat data. We prove that the corresponding solutions are global to the past and the future. This is the first small-data global-in-spacetime existence result for any quasilinear equation on $\mathbb{R}^3$ that fails to satisfy the null condition. In both theorems, the techniques we use are robust and applicable to a wide class of quasilinear hyperbolic PDE.