01
The problem
Resolve global existence and smoothness for the three-dimensional incompressible Navier–Stokes equations under the official conditions, or exhibit finite-time breakdown.
Date noteThe equations date to the 1820s–1840s; 1934 marks Leray’s modern 3D regularity question. The exact Clay formulation is from 2000.
Open since1934historical modern form
Last checked2026.07.26Catalog verification
02
Reward offers
Offer 01$1,000,000Clay Mathematics Institute
InstitutionalClay does not accept direct submissions. A proposed solution must be published in a qualifying outlet, at least two years must pass, and the work must gain general acceptance in the mathematics community while resolving the complete official formulation.