Finite-time blowup for smoothly forced three-dimensional Euler

Can a finite-energy solution of the three-dimensional incompressible Euler equations on R³ develop unbounded vorticity from smooth compactly supported initial data when the external forcing remains smooth through the blowup time?

Why it matters and what counts as progress

Why it matters: Smooth forcing removes a regularity limitation in earlier forced-blowup constructions.

What counts as progress: A finite-time blowup construction with the forcing smooth through the terminal time and the claimed preterminal uniqueness and vorticity behavior proved; unforced blowup is a separate target.

How the problem developed

2023 · Partial result [1]
Diego Córdoba and Luis Martínez-Zoroa Developed a forced three-dimensional Euler blowup construction with force regularity C^{1,1/2−ε}; it did not reach a force smooth in space and time.
8 Sept 2026 · Reported claim [2][3]
Levent Alpöge and Tristan Buckmaster Reported finite-time vorticity blowup for smooth initial data under a force smooth in space and time through blowup; the paper reports Lean verification and AI assistance.

AI contribution

Blowup for the Euler Equations with Smooth Forcing

Alpöge and Buckmaster construct smooth compactly supported initial velocity and forcing smooth through blowup time for 3D incompressible Euler on R³. A finite-energy classical solution develops unbounded vorticity in finite time, with divergent time-integrated vorticity sup norm and uniqueness in the stated preterminal class.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 with Mathlib, Comparator and exact interval-arithmetic certificates)

Human understandingPartial