Conjecture Fieldbook

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Navier–Stokes existence

Canonical problem dossier

For three-dimensional incompressible flow, do smooth solutions always exist for all time, or can a singularity form?

Status
Open
Field
Analysis
Verification
Reviewed dossier
Source grade
mixed
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

∂tu+(u⋅∇)u=−∇p+νΔu,∇⋅u=0\partial_t u+(u\cdot\nabla)u=-\nabla p+\nu\Delta u,\qquad \nabla\cdot u=0

Why it matters

The equations underpin fluid mechanics. A proof would clarify whether their mathematical model can remain smooth or break down from smooth initial conditions.

Context and known approaches

Weak solutions exist in three dimensions, but global regularity and uniqueness for smooth data remain unresolved.

Formal boundary

Prove global existence and smoothness for admissible three-dimensional initial data, or construct a finite-time breakdown satisfying the official conditions.

Known partial results

  • Global weak solutions are known in three dimensions; smoothness and uniqueness remain the stated gap.

Source trail

  1. Clay Mathematics Institute — Official Millennium problem overview.
  2. Wikipedia — Problem background and references.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.