MATHPH In the literature: contested

Finite-time blowup of 3D Euler from smooth data without boundaries

In plain words

For an ideal fluid with no viscosity, it is unknown whether a smooth flow filling all of space can concentrate its rotation into infinite values in finite time. Blowup is proven when the fluid touches a wall or when the starting flow is slightly rough.

Precise statement

Incompressible Euler on $R^3$ (or $T^3$): does there exist smooth finite-energy initial data (e.g. $C_c^{\infty}$) whose solution satisfies $\operatorname{integral}_0^T \mid\mid \omega(t)\mid\mid_{\operatorname{inf}} dt = \infty$ at a finite $T$, with $\omega = \operatorname{curl} u$ (Beale-Kato-Majda criterion)? Known: blowup for smooth data in a cylinder with boundary (Chen and Hou, computer-assisted, PNAS 2025) and for $C^{(1,\alpha)}$ data on $R^3$ (Elgindi, Annals of Mathematics 2021, small $\alpha$; Cordoba, Martinez-Zoroa and Zheng, Annals of PDE 2025). Answer: a verified construction or a proof of global regularity.

What would settle it

An independently verified proof, most likely a computer-assisted nonlinear stability estimate around an approximate self-similar profile.

Status in the literature

Unverified note

Programmatic searches in September 2026 claimed two Euler singularities (as reported by Petrillo and Glimm, arXiv 2609.23868), and Ganeshram, Duruisseaux and Anandkumar (arXiv 2609.10867) posted a candidate self-similar profile on $R^{3}$ at rate $1/2$ with a stability framework but no completed proof; none is independently verified.