Breakthroughs Weekly — 8 September 2026
A finite-time blowup proof for forced 3D Euler flows earns Terence Tao's praise as a Navier-Stokes stepping stone, and a five-decade percolation conjecture falls.
Mathematicians prove finite-time blowup for smoothly forced 3D Euler, Boussinesq and porous-medium flows
NYU · 7 Sep 2026 · Partial differential equations
Levent Alpöge (Anthropic) and Tristan Buckmaster (NYU) constructed smooth initial data and smooth, compactly supported forcing under which solutions to the three-dimensional incompressible Euler equations, the two-dimensional Boussinesq system, and the incompressible porous-medium equation all become singular in finite time, extending a multiscale layering construction pioneered by Diego Córdoba and Luis Martínez-Zoroa. The Euler and porous-medium proofs were formally checked in the Lean proof assistant, and the authors say large language models, including Claude and OpenAI’s models, assisted heavily in developing the arguments. Fields medalist Terence Tao, who had no role in the work, called it “a remarkable achievement” and a genuine stepping stone toward the far harder unforced blowup problem for Navier-Stokes, while noting substantial technical obstacles remain before that Millennium Prize problem itself is reached. A separate, unpublished claim of an AI-generated proof for forced Navier-Stokes from OpenAI has not been made available for independent scrutiny and is not covered here.
A five-decade percolation puzzle falls, completing a program begun by Benjamini and Schramm
arXiv · 3 Mar 2026 · Probability theory
Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, and Vincent Tassion proved that on any infinite transitive graph, past the critical probability for percolation, the chance a given point lies in a large finite cluster decays exponentially in the graph’s isoperimetric profile. The result settles the “supercritical” half of a sharpness conjecture whose “subcritical” half was solved in 2007, closing out a research program Itai Benjamini and Oded Schramm proposed decades ago, using methods described as strikingly simple for how broadly they apply. The preprint drew little notice outside probability circles until Quanta profiled it on 31 August 2026, where probabilists including Tel Aviv University’s Asaf Nachmias praised the proof as stunning.