In three space dimensions, given smooth initial data for the incompressible Navier–Stokes equations with periodic boundary conditions (or suitable decay), prove that smooth solutions exist for all time — or produce a singularity forming in finite time.
The Clay Millennium formulation (Fefferman, 2000) crystallised a question implicit since the 19th century. Two-dimensional smoothness is classical (Ladyzhenskaya); in 3D, weak global solutions exist (Leray/Hopf) but their uniqueness and regularity are open. Blow-up candidates (Kiselev–Šverák, Hou's scenarios) keep failing just past the point of provability.
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