Do smooth finite-energy solutions of the incompressible porous media equation 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.)
Incompressible porous media flow (Darcy's law with incompressibility) sits between 2D Euler and the SQG equation in the family of active scalars; Córdoba–Faraco–Gancedo (2011) showed the unstable (heavier-on-top) case can turn over in finite time. Global regularity of the stable case and general blow-up stayed open. In September 2026 Alpöge–Buckmaster announced finite-time blow-up with smooth forcing (preprint, Lean-formalized) via the Córdoba–Martínez-Zoroa iteration. The unforced case remains open.
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