Global uniqueness of maximal globally hyperbolic developments for shock-forming data

Prove global uniqueness of maximal globally hyperbolic developments for shock-forming initial data for the compressible Euler equations, including uniqueness beyond the compact portions of the development currently constructed.

Background

The paper and its predecessor construct portions of the classical development up to parts of the singular boundary and Cauchy horizon. These portions are extendible and therefore do not themselves determine a unique maximal globally hyperbolic development.

The authors explain that known sufficient criteria for uniqueness require constructing the entire maximal globally hyperbolic development and establishing a global one-sidedness property along its boundary. Thus, the local constructions in the paper do not resolve global uniqueness.

References

(Non-unique MGHDs). Is there any smooth data for the compressible Euler equations that yields two distinct, intersecting globally hyperbolic developments $\mathbf{M}_1$ and $\mathbf{M}_2$ such that $\mathbf{M}_1 \neq \mathbf{M}_2$, and such that the corresponding solutions disagree at some point $p \in \mathbf{M}_1 \cap \mathbf{M}_2$?

— Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime  (2609.08158 - Abbrescia et al., 8 Sep 2026) in Section 1, Introduction, discussion immediately preceding Section 1.1; paragraph titled “The above discussion suggests the following open problems”

In any spatial dimension, uniqueness of MGHDs for shock-forming data remains an open problem.

— The emergence of the Cauchy horizon from the crease for $3D$ compressible Euler flow  (2609.03101 - Abbrescia et al., 2 Sep 2026) in Section 1, subsection “A rough overview of our work”; also Section 1, subsection “The breakdown of classical determinism -- a fundamental open problem”

It is not clear whether the fluid can be followed up to Cauchy horizon from data on $\Sigma_0$ without first constructing $$ as we did in .

— The emergence of the Cauchy horizon from the crease for $3D$ compressible Euler flow  (2609.03101 - Abbrescia et al., 2 Sep 2026) in Section 1, subsection “The Cauchy initial value problem for the Euler equations”