On classical solutions and singularity formation in incompressible fluids

Guest post by Diego Córdoba and Luis Martínez-Zoroa on Terence Tao's blog, dated 4 Oct 2026 (Tao's note: the post "was initially written in a different file format and converted using AI"). It is an expository survey of their programme to build finite-time singularities for incompressible fluid equations inside regimes where classical local existence and uniqueness hold. The survey covers instantaneous loss of regularity (Bourgain–Li, Elgindi–Masmoudi, their own work with Ożański), Elgindi's C^{1,α} Euler blow-up and recent sub-1/3 results (Shkoller, Chen, Shao–Wei–Zhang–Zhang), and their "cascade of vortex layers" construction: vorticity on widely separated scales, where each layer amplifies the next and the stage times sum to a finite blow-up time.

Results they list: forced 3D Euler blow-up with velocity in C^{3,1/2}∩L² and force in C^{1,1/2−ε}∩L²; blow-up for fractional (hypodissipative) Navier–Stokes for 0 < α < (22−8√7)/9 ≈ 0.093 (with Fan Zheng), with force in L¹_t C^{1,ε}_x; IPM blow-up with a source smooth in space; and Boussinesq, variable-density Euler and gSQG extensions. They stress that their dissipative result "concerns small fractional powers of the Laplacian, not ordinary Navier–Stokes", and that the ordinary Laplacian cannot be reached by rescaling the construction.

On the AI-era results: Alpöge, Buckmaster and Coiculescu extended the IPM construction to a source smooth in space and time, and the Alpöge–Buckmaster Boussinesq and Euler manuscripts "explicitly build on amplification through successive layers". They describe OpenAI's 8 Sep 2026 announcement as "two separate claims: blow-up for ordinary Navier–Stokes with smooth forcing, and blow-up for unforced Euler from smooth initial data". The first, "if verified under the stated hypotheses, would establish an allowed breakdown alternative in the Millennium Problem formulation. It is distinct from unforced Navier–Stokes blow-up." They add: "At this moment we are not in a position to say anything substantive about a comparison between OpenAI's constructions and our cascade methods."

Reach was low: the Hacker News submission (item 49954713) had 1 point at the time of checking.

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