Navier–Stokes existence and smoothness
Do smooth solutions of the 3D Navier–Stokes equations always exist?
Summary
Asks whether smooth, globally defined solutions always exist for the three-dimensional equations of fluid flow, or whether they can break down.
Source
Open, as listed by Clay Mathematics Institute · Millennium Prize Problems. Checked 29 Sep 2026.
The status is the source’s. Papex does not decide whether a problem is solved.
Prize
Clay Mathematics Institute · $1,000,000
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