Existence and uniqueness of maximal globally hyperbolic developments for compressible Euler flow

Establish the existence and uniqueness of a largest possible classical solution, or maximal globally hyperbolic development, for shock-forming solutions of the compressible Euler equations arising from smooth initial data.

Background

The paper constructs a bounded portion of the boundary of a maximal globally hyperbolic development for three-dimensional compressible Euler solutions with dynamic entropy and vorticity, including a crease, a singular boundary, and a Cauchy horizon. However, the constructed region is only local in spacetime and does not constitute the entire maximal globally hyperbolic development.

Global existence and uniqueness require controlling the totality of the classical solution region and proving suitable global boundary properties. The authors emphasize that this issue is substantially more delicate than local uniqueness and remains unresolved for shock-forming compressible Euler flow in every spatial dimension.

References

We again highlight that it remains a fundamental outstanding open problem of the field to prove existence and uniqueness of a largest possible classical solution.

— 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 breakdown of classical determinism -- a fundamental open problem”; see also Section 1, subsection “MGHDs”

While the shock development problem is still open in $3D$ in full generality, there has been exciting progress, which we described in Sect. 1.9.12.

— 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 less rough overview of our work and methods”

Without transversal convexity, a continuum of characteristics could collapse into the same co-dimension $2$ set, and if that happens, it is not clear how to construct $$ or whether the shock development problem is well-posed.

— 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 less rough overview of our work and methods”