Can flow through porous rock focus itself into a singularity? Forced blow-up claimed September 2026 (Alpöge–Buckmaster, preprint + Lean); unforced stays open.
The Problems
Not schoolwork — the questions that resisted Erdős, Hilbert, and everyone since. The club keeps three shelves: what is still open, what AI recently settled, and what took humanity centuries.
Tagged fluid-dynamics — 5 entries. Clear
Can buoyancy-driven flow focus itself into a singularity? The forced case was claimed in September 2026 (Alpöge–Buckmaster, preprint + Lean); unforced stays open.
Can a perfectly smooth ideal fluid focus itself into a singularity in finite time? Known for rough data since 2021; the smooth case would hand Navier–Stokes a blow-up blueprint.
Onsager's conjecture
historic
Lars Onsager predicted in 1949 that rough fluid flows can dissipate energy without viscosity below Hölder exponent 1/3, but not above. Proved in full by 2018 via convex integration.
Do solutions of the equations that describe fluid motion always exist and stay smooth, or can they blow up? Mathematics has not caught up with the water in the glass.
pdes fluid-dynamics millennium-prize mathematical-physics blow-up regularity