Do smooth finite-energy solutions of the three-dimensional Boussinesq system stay regular for all time, or can they develop a singularity in finite time? (Known: blow-up with smooth forcing announced September 2026, preprint; the unforced case is open.)
The Boussinesq system couples incompressible flow to a transported temperature (or density) field — the standard model for buoyancy-driven motion. Its 2D regularity theory advanced through Chae (2006) and Hmidi–Keraani (2007), while the 3D case stayed open. The Córdoba–Martínez-Zoroa iteration gave a new route: in September 2026 Alpöge–Buckmaster announced finite-time blow-up with smooth forcing (preprint, Lean-formalized, 76 pages for the Boussinesq case — the model setting where the amplitude-frequency ODE is simplest). The unforced case remains open.
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