Finite-time blowup of 3D Euler from smooth data without boundaries
This text mentions work from 2024 on. Those parts were partly written without a live literature search and have not been checked by an editor yet.
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 $\int_0^T \mid\mid \omega(t)\mid\mid_{\operatorname{inf}} dt = \infty$ 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: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
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.
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See also
- Related Global regularity or blowup of unforced 3D Navier-Stokes flows
- Related Finite-time singularity of the inviscid surface quasi-geostrophic equation
- Related Rigorous unstable self-similar singularities in incompressible fluid models
- Related Is the dissipation anomaly strong or weak in wall-free turbulence?
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