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Claim under reviewAnalysis & fluids · Calculus → research

The Navier–Stokes question

Can smooth fluid motion break down?

Reviewed: 2026-10-20Editorial introduction · not original research
∂tu+(u⋅∇)u=−∇p+νΔu+f,∇⋅u=0\partial_t u+(u\cdot\nabla)u=-\nabla p+\nu\Delta u+f,\qquad\nabla\cdot u=0

01 / The question

These equations describe velocity, pressure and viscosity in a fluid. The famous problem concerns the mathematical behavior of three-dimensional incompressible flow, including whether smooth solutions can develop singularities.

02 / Why it is hard

A realistic-looking simulation only samples motion on a finite grid. It cannot alone establish existence or regularity for every admissible initial condition.

03 / Progress & scope

OpenAI announced a proof for a forced singularity formulation on September 8, 2026, with Lean artifacts. Clay currently labels its page Active. We present the announcement and institutional status separately, without declaring prize recognition.

Try an experiment

Make an abstract question tangible

The vector field below is an illustrative divergence-free field, not a numerical Navier–Stokes solver or a rendering of the proposed singularity.

u(x,y)=A(sin⁡xcos⁡y,−cos⁡xsin⁡y),∇⋅u=0u(x,y)=A(\sin x\cos y,-\cos x\sin y),\quad\nabla\cdot u=0

Arrows show local direction and relative magnitude. The slider changes amplitude A only; there is no time evolution or 3D singularity simulation. Differentiate the two components to see why their divergence cancels.

Sources & further reading

Clay Mathematics InstituteOpenAI · September 8, 2026

Sources last checked 2026-10-20. Status reflects an editorial review on that date, not continuous live monitoring.

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