Tristan Buckmaster Constructs Finite-Time Blowup for 3D Euler with Smooth Forcing

September 14, 2027

On September 7, Tristan Buckmaster, Professor of Mathematics at Courant Institute School of Mathematics, Computing, and Data Science, together with Levent Alpöge, a mathematician at Anthropic, announced the construction of solutions to the three-dimensional incompressible Euler equations, which start from smooth data of finite energy, are subject to a force that is smooth in space and in time, and lose regularity in finite time. Two companion papers do the same for the inviscid Boussinesq system in the plane and, for the incompressible porous media equation on the two-dimensional torus, jointly with Matei P. Coiculescu. The results in all three papers have been formalized and verified using Lean proof assistant.

This is an outstanding achievement and represents a significant step towards the resolution of the Millennium problem for 3D Navier-Stokes.

The Euler construction is axisymmetric with swirl, supported in a fixed solid torus. Circulation and meridional velocity stay bounded, while the gradient of the circulation and the vorticity both become infinite in the supremum norm, and the time integral of the supremum of the vorticity diverges. Before the blowup time the solution is unique among locally space-time Lipschitz solutions of finite energy, and the competitors need not be axisymmetric. The proof strategy was initiated by Diego Córdoba (ICMAT, Madrid) and Luis Martínez-Zoroa (CUNEF University), who considered "rough" forces in the context of 3D Euler. In contrast, for the work of Buckmaster and collaborators, the force is compactly supported and smooth up to and including the blowup time.

The proofs were obtained with the assistance of large language models, as detailed in the papers, and verified using the Lean proof assistant.

Writing on September 7, Terence Tao (UCLA) called the work "a remarkable achievement", and said that the authors "have managed to push one of the major promising approaches towards constructing blowup solutions to fluid equations"—as developed by Córdoba and Martínez-Zoroa—“to establish finite time blowup for many key fluid equations, including 3D incompressible Euler, with a smooth forcing term." He added that nothing in principle appears to prevent the method from reaching Navier-Stokes.

Buckmaster has been a Professor of Mathematics at NYU since 2023. He received his doctorate from the University of Leipzig in 2014 and was a Courant Instructor from 2014 to 2017, an Assistant Professor at Princeton from 2017 to 2022, and a Professor at the University of Maryland from 2022 to 2023. He received the Clay Research Award in 2019 and delivered the Hadamard Lectures at the IHÉS in 2020. Alpöge is a mathematician at Anthropic, while Coiculescu is a first year Courant Instructor at NYU.

Let us congratulate Tristan and collaborators on this incredible achievement!