PPL 119 / 177 permanent IDsPrize Problem Ledger · PPLChecked 2026.07.26
Prize Problem LedgerInstitutionalPPL 119Navier–Stokes existence and smoothness

Permanent problem IDPPL 119

Verified openInstitutionalexistence

Partial differential equations

Navier–Stokes existence and smoothness

Resolve global existence and smoothness for the three-dimensional incompressible Navier–Stokes equations under the official conditions, or exhibit finite-time breakdown.

fluid dynamicsanalysismathematical physics
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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
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Reward offers

Offer 01$1,000,000Clay Mathematics Institute
Institutional

Clay 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.

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Sources & reading