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The official problem specifies alternatives A/B for unforced regularity and C/D for forced breakdown, with separate decay, periodicity and energy conditions.
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Leray showed that the three-dimensional equations always have a weak solution with suitable growth properties; uniqueness of weak solutions is not known.
Passage ID: navier-stokes:clay-leray:v1
Source item: clay-leray
Source record: navier-stokes:source:clay:v1
Caffarelli–Kohn–Nirenberg (simplified by Lin): the singular set E of a suitable weak solution has P¹(E) = 0, and such solutions exist for the given data.
Passage ID: navier-stokes:clay-partial-regularity:v1
Source item: clay-partial-regularity
Source record: navier-stokes:source:clay:v1
For data not assumed small, (A) and (B) hold on a short time interval depending on the data; for ν > 0 a finite blowup time forces the velocity to become unbounded.
Passage ID: navier-stokes:clay-short-time:v1
Source item: clay-short-time
Source record: navier-stokes:source:clay:v1
In three dimensions, (A) and (B) hold provided the initial velocity satisfies a smallness condition.
Passage ID: navier-stokes:clay-small-data:v1
Source item: clay-small-data
Source record: navier-stokes:source:clay:v1
In two dimensions the analogues of assertions (A) and (B) have long been known (Ladyzhenskaya), also for the Euler equations; the description notes this gives no hint about three dimensions.
Passage ID: navier-stokes:clay-two-dimensions:v1
Source item: clay-two-dimensions
Source record: navier-stokes:source:clay:v1
The official problem specifies alternatives A/B for unforced regularity and C/D for forced breakdown, with separate decay, periodicity and energy conditions.
Passage ID: navier-stokes:entry:clay:v1
Source item: clay
Source record: navier-stokes:source:clay:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v8
Source assessment
Proved
Established result reported in the cited source
Only the exact statement and source passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Established result reported in the cited source
Only the exact statement and source passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Established result reported in the cited source
Only the exact statement and source passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Established result reported in the cited source
Only the exact statement and source passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Proved
Established result reported in the cited source
Only the exact statement and source passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Source assessment
Open
Unforced periodic regularity target
Alternative B; the selected forced construction does not settle this assertion.
Observed Oct 13, 2026
Scope
Unforced global regularity on the three-torus
Source assessment
Open
Unforced whole-space regularity target
Alternative A; the selected forced construction does not settle this assertion.
Observed Oct 13, 2026
Scope
Unforced global regularity in the whole space
Question
Navier–Stokes existence and smoothness
Research activity
Compare manuscript, comparator statement and proof dependencies
Check the official decay/periodic predicates, force regularity, energy condition and imported proof adapters at the pinned theorems.
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History
Four official existence/breakdown alternatives
Further reading
Consult the original question and its hypotheses.
Recorded authorship
Charles L. Fefferman
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Existence and Smoothness of the Navier–Stokes Equation
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