Do there exist smooth, finite-energy initial data for the three-dimensional incompressible Euler equations whose solution develops a singularity in finite time? (Known: yes for \(C^{1,\alpha}\) data (Elgindi 2021); the smooth case is open.)
The incompressible Euler equations are Navier–Stokes with zero viscosity, so any Euler blow-up scenario is a candidate mechanism for Navier–Stokes blow-up too. Beale–Kato–Majda (1984) showed blow-up requires unbounded vorticity. Hou–Luo (2013) found a numerically compelling potentially-singular axisymmetric scenario at the boundary. Elgindi (2021) proved the first finite-time, asymptotically self-similar blow-up — but for \(C^{1,\alpha}\) data, not smooth data. In September 2026 Alpöge–Buckmaster announced finite-time blow-up for 3D Euler with smooth forcing (preprint, Lean-formalized, via the Córdoba–Martínez-Zoroa iteration); days later OpenAI announced a claimed unforced smooth Euler blow-up (preprint + Lean, ~50 hours of agent computation) — which, if verified, would close this entry. A numerically stable unforced blow-up ansatz via physics-informed neural networks was reported the same week, without a rigorous stability proof. All September 2026 claims await independent verification; the entry stays open pending review.
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