Petrillo and Glimm Define a Precise Target for Unforced 3D Navier‑Stokes Blowup Searches

A new arXiv preprint is not claiming to solve the Navier-Stokes millennium problem. Instead, it tries to pin down the next concrete target for anyone hoping to prove finite-time blowup in the part that still appears open: the unforced three-dimensional equations.

In “The Positive Defect Problem: Target and Admissibility Criteria for a Programmatic Search for Unforced Navier-Stokes Blowup,” posted as arXiv:2609.23868, Jarret Petrillo and James Glimm propose a mathematically precise search target and argue that any credible computer-assisted attack on the unforced case should be organized around it.

That matters because the burst of September 2026 activity around fluid singularities largely centered elsewhere. As Petrillo and Glimm put it, “On 7–8 September 2026 programmatic search produced singularities …” in forced Navier-Stokes and in Euler. But the unforced 3D Navier-Stokes case — the version many mathematicians regard as the harder remaining frontier — has not been resolved.

In plain language, the paper asks whether a standard weak solution of the unforced equations, starting from smooth initial data on a periodic box, can lose energy over a finite time window by more than ordinary viscosity should allow at a fixed viscosity. The authors call that the positive defect problem.

That formulation is narrower than a full blowup claim, but it is the point of the paper. Petrillo and Glimm argue that a positive defect would imply blowup, which in turn would give a negative answer to statement (B) in the Clay Mathematics Institute’s formulation of the Navier-Stokes existence and smoothness problem. They also stress a limit: the converse is not known, so the target is a one-way implication, not a full equivalence to singularity.

The paper says this target can also be expressed on the Fourier side of the equations, as a lower bound averaged over time on energy flux through Littlewood-Paley shells. For non-specialists, the key point is that the authors are translating the question into a measurable threshold for how energy must cascade across scales, rather than offering a free-form numerical hunt.

They also spell out several necessary conditions that any candidate blowup would have to satisfy. Among them, the paper says, are a Type II singular time in the velocity field and a failure to remain in the Onsager-critical regularity class, a threshold tied to how rough a turbulent flow can become while still conserving energy.

Much of the reduction is formalized in Lean 4, a proof assistant used to check mathematical arguments step by step. “Every implication not marked otherwise is a theorem in Lean 4 over Mathlib,” the paper says. The authors add an important qualifier: Mathlib has no native Navier-Stokes object, so the equation enters only through hypotheses. In other words, the formalization checks the logical structure of the reductions, not a built-in fluid-dynamics theory.

The paper is equally explicit about what would not be enough. It says “no finite computation” can by itself certify the Galerkin-uniform ceiling needed for the argument. And it says that adding forcing or a selection rule does not evade the issue; it simply pushes the problem back to the same positive-defect target.

To give a sense of scale, Petrillo and Glimm report 12 pseudo-spectral simulations at resolutions of 128 cubed and 256 cubed. In those runs, they say, the “fine-shell flux floor” stayed within 1% to 6% of the required ceiling for about one-third of a turnover time, but failed in scale at the Kolmogorov wavenumber, the small-scale cutoff where viscosity usually dominates. Their interpretation is that a genuine unforced blowup candidate would need to differ from ordinary turbulence not mainly in timing, but in how deeply the energy cascade penetrates to finer scales.

That conclusion is the paper’s central contribution: not a proof, but a map of what a proof would likely have to show.

The backdrop is unusually charged. OpenAI posted a manuscript on Sept. 8 claiming a finite-time singularity for forced 3D Navier-Stokes, and Levent Alpöge and Tristan Buckmaster posted an Euler preprint on Sept. 7; both remain under community review. On Sept. 11, Clay said the Navier-Stokes problem “has apparently been settled,” while also making clear that its prize review is deliberate and ongoing.

Petrillo and Glimm’s paper does not settle that process. It is an arXiv preprint, not a peer-reviewed result, and it does not claim a proof of unforced Navier-Stokes blowup. What it does claim is more specific, and potentially useful: a formalized set of criteria for what the still-open target would have to look like if anyone is going to find it.