{"id":"navier-stokes-existence-and-smoothness","domain":"mathematical-physics","type":"Open Problem","title":"Navier-Stokes Existence and Smoothness","slug":"navier-stokes-existence-and-smoothness","url":"/mathematics/mathematical-physics/navier-stokes-existence-and-smoothness/","posed":"2000 (formal statement), Charles Fefferman","status":"open","summary":"Do smooth solutions to 3D Navier-Stokes fluid equations always exist for all time? Clay Millennium Prize. 2D case solved; 3D remains open despite everyday engineering use.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Fefferman, C.L. (2000). Existence and smoothness of the Navier-Stokes equation. Clay Mathematics Institute."},{"tier":1,"citation":"Caffarelli, L., Kohn, R. and Nirenberg, L. (1982). Partial regularity of suitable weak solutions. Communications on Pure and Applied Mathematics, 35(6), 771-831."},{"tier":2,"citation":"Tao, T. (2016). Finite time blowup for an averaged three-dimensional Navier-Stokes equation. Journal of the American Mathematical Society, 29(3), 601-674."},{"tier":3,"citation":"Devlin, K. (2002). The Millennium Problems. Basic Books."}],"relationships":[{"type":"built_on","target":"differential-equation","label":"Differential Equation"},{"type":"related_to","target":"clay-millennium-problems","label":"Clay Millennium Problems"},{"type":"tracked_in","target":"proof-status","label":"Proof Status Board"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.3"}