Math Conjectures

Three-dimensional Navier–Stokes existence and smoothness

OPENIconicOpen problemProposed 1934 · Canonical special case

Canonical statement

Let u0:R3→R3u_0:\mathbb R^3\to\mathbb R^3 be C∞C^\infty, divergence-free, and satisfy sup⁡x(1+∣x∣)N∣∂αu0(x)∣<∞\sup_x(1+|x|)^N|\partial^\alpha u_0(x)|<\infty for every NN and multi-index α\alpha. There exist u∈C∞(R3×[0,∞);R3)u\in C^\infty(\mathbb R^3\times[0,\infty);\mathbb R^3) and p∈C∞(R3×[0,∞))p\in C^\infty(\mathbb R^3\times[0,\infty)) satisfying
∂tu+(u⋅∇)u=−∇p+Δu,∇⋅u=0,u(⋅,0)=u0 \partial_tu+(u\cdot\nabla)u=-\nabla p+\Delta u,\qquad \nabla\cdot u=0,\qquad u(\cdot,0)=u_0
for all t≥0t\ge0, with sup⁡0≤t≤T∫R3∣u(x,t)∣2 dx<∞\sup_{0\le t\le T}\int_{\mathbb R^3}|u(x,t)|^2\,dx<\infty for every T<∞T<\infty.
View source LaTeX
Let \(u_0:\mathbb R^3\to\mathbb R^3\) be \(C^\infty\), divergence-free, and satisfy \(\sup_x(1+|x|)^N|\partial^\alpha u_0(x)|<\infty\) for every \(N\) and multi-index \(\alpha\). There exist \(u\in C^\infty(\mathbb R^3\times[0,\infty);\mathbb R^3)\) and \(p\in C^\infty(\mathbb R^3\times[0,\infty))\) satisfying \[ \partial_tu+(u\cdot\nabla)u=-\nabla p+\Delta u,\qquad \nabla\cdot u=0,\qquad u(\cdot,0)=u_0 \] for all \(t\ge0\), with \(\sup_{0\le t\le T}\int_{\mathbb R^3}|u(x,t)|^2\,dx<\infty\) for every \(T<\infty\).
PDE-001:[Leray1934NavierStokes][Fefferman2006NavierStokes][Clay2026NavierStokes]
Partial differential equationsMathematical physics

Notes

The problem asks whether every smooth, rapidly decaying, divergence-free initial velocity field on R3\mathbb R^3 launches a globally smooth solution of the incompressible Navier–Stokes equations: a velocity uu and pressure pp, smooth for all t≥0t\ge 0, with kinetic energy bounded on every bounded time interval. The question is conventionally dated to Leray's 1934 memoir [Leray1934NavierStokes], which constructed global weak solutions and left their regularity undecided; the precise modern formulation is that of the Clay Millennium Prize problem [Fefferman2006NavierStokes] [Clay2026NavierStokes].

What is known sits on either side of the gap. Leray–Hopf weak solutions exist globally for finite-energy data [Leray1934NavierStokes], smooth solutions exist at least locally in time, and a large body of conditional criteria guarantees regularity under additional integrability or smallness hypotheses [Fefferman2006NavierStokes]. None of this decides whether weak solutions must remain smooth.

The problem remains open: a resolution requires either a proof that arbitrary smooth finite-energy data never develop a finite-time singularity, or an explicit smooth blow-up example.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.