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navier-stokes
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The equations predict every weather forecast and airfoil, yet nobody knows whether their solutions can blow up in finite time. In two dimensions they are known to stay smooth. Tao showed in 2016 that a modified, averaged version can blow up, so any proof must use structure that the averaged equation lacks. Recent computer-assisted work on the related Euler equations has produced rigorous blow-up scenarios.
Posed 1822–1845, Navier and Stokes wrote the equations; the Clay form dates to 2000.
What would settle it
Lean statement in formal-conjectures
Prize
$1,000,000 Clay Millennium Prize
Related problems
- The original curation links this problem to Turbulence; the argument for the link has not been written yet.