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For the three-dimensional incompressible Navier–Stokes equations in R3\mathbb{R}^3, it is unknown whether smooth finite-energy axisymmetric initial data with nonzero swirl (nonzero angular velocity component) generate solutions that remain smooth for all time, or whether some such data produce a finite-time singularity. All unconditional results either remove the swirl (a case that is essentially two-dimensional and globally regular) or impose additional critical smallness, decay, or form-boundedness conditions on the swirl quantity Γ=rvθ\Gamma=r v^{\theta}, which is bounded by a maximum principle but not controlled sharply enough near the symmetry axis. This is the canonical structured open subproblem of the Clay Navier–Stokes millennium problem: a finite-time blowup within this class would resolve the millennium problem negatively for R3\mathbb{R}^3, while a global regularity proof would be the first large-data global regularity result beyond the essentially two-dimensional regime. The problem remains open in the specialist literature through 2026.

Background

The incompressible Navier–Stokes equations govern the velocity field v(x,t)v(x,t) and pressure p(x,t)p(x,t) of a viscous incompressible fluid through ∂tv+(v⋅∇)v−Δv+∇p=0\partial_t v+(v\cdot\nabla)v-\Delta v+\nabla p=0 and ∇⋅v=0\nabla\cdot v=0 in R3×(0,∞)\mathbb{R}^3\times(0,\infty). Leray–Hopf weak solutions (distributional solutions satisfying the global energy inequality) exist globally for finite-energy data, but their smoothness and uniqueness are unknown; this is the Clay Mathematics Institute millennium problem of existence and smoothness versus breakdown. A flow is axisymmetric when, in cylindrical coordinates (r,θ,z)(r,\theta,z), the components (vr,vθ,v3)(v^r,v^{\theta},v^3) depend only on (r,z)(r,z); the angular component vθv^{\theta} is called the swirl, and the flow is swirl-free when vθ≡0v^{\theta}\equiv 0. The vorticity ω=∇×v\omega=\nabla\times v satisfies a transport-diffusion equation whose nonlinear vortex-stretching term ω⋅∇v\omega\cdot\nabla v, absent in two dimensions, is the principal mechanism that can amplify vorticity and is the core obstacle to regularity. The equations are invariant under the scaling vλ(t,x)=λv(λ2t,λx)v_\lambda(t,x)=\lambda v(\lambda^2 t,\lambda x); an a priori estimate is critical when it is invariant under this scaling, and supercritical when it is weaker, since supercritical bounds cannot control the solution at small scales. The kinetic-energy bound is supercritical for the full 3D problem.

For axisymmetric flows without swirl, Ukhovskii and Iudovich (1968) proved global existence of smooth solutions for all finite-energy data, because the system then reduces essentially to a two-dimensional problem without vortex stretching. With nonzero swirl the problem retains genuine three-dimensional vortex stretching concentrated near the symmetry axis, and global regularity is open. The key a priori structure is that the quantity Γ=rvθ\Gamma=r v^{\theta} satisfies a linear transport-diffusion equation with no zeroth-order term, hence obeys the maximum principle sup⁡t≥0∥Γ(t)∥L∞≤∥Γ(0)∥L∞\sup_{t\ge 0}\|\Gamma(t)\|_{L^\infty}\le\|\Gamma(0)\|_{L^\infty}; however, this bound is insufficient by itself for regularity, since vθ=Γ/rv^{\theta}=\Gamma/r may still grow like 1/r1/r toward the axis.

Around 2008–2009, Chen–Strain–Yau–Tsai and, independently, Koch–Nadirashvili–Seregin–Šverák proved that if ∣v(x,t)∣≤C/r|v(x,t)|\le C/r uniformly then the solution is smooth for all time, so that any blowup, if it exists, must have rate faster than 1/r1/r; the proofs rest on Liouville theorems for ancient solutions (solutions extending backward to time −∞-\infty) obtained by rescaling a hypothetical singularity. Jiu–Xin and others obtained global regularity under smallness of suitable scale-invariant (zero-dimensional, critical) norms of the swirl.

The source paper underlying this problem, Lei and Zhang (arXiv:1505.02628; Pacific Journal of Mathematics 289 (2017) 169–187), made two contributions. First, they observed that the axisymmetric vorticity system for (ωr/r, ωθ/r, ω3)(\omega^r/r,\,\omega^{\theta}/r,\,\omega^3), treating ωr/r\omega^r/r and ω3\omega^3 as potentials generated by Γ\Gamma, is critical rather than supercritical under the natural scaling, so the scaling gap for the axisymmetric problem is in this sense only logarithmic. Second, they proved that all Leray–Hopf solutions with axisymmetric initial data in H1/2H^{1/2} with Γ0∈L∞\Gamma_0\in L^\infty are globally smooth provided vθv^{\theta} satisfies a scale-invariant Form Boundedness Condition (a weighted one-dimensional L2L^2 coercivity condition on vθ/rv^{\theta}/r), which is implied by the pointwise decay ∣Γ∣≤C∗∣ln⁡r∣−2|\Gamma|\le C_*|\ln r|^{-2} near the axis; they also proved global regularity when ∥Γ0∥L∞\|\Gamma_0\|_{L^\infty} or the local-in-rr supremum of ∥Γ∥L∞\|\Gamma\|_{L^\infty} is small, with the smallness threshold depending on a dimensionless quantity of the data. Subsequent work (surveyed by Zhang and Pan, Analysis in Theory and Applications 38 (2022) 243–296, revised on arXiv in August 2024) produced further critical and slightly supercritical regularity criteria, including logarithmic improvements, but none removes the need for an additional critical hypothesis. Numerically, Hou (Foundations of Computational Mathematics 23 (2023), no. 6, 2251–2299) found potentially singular behavior for axisymmetric Navier–Stokes with swirl near the origin, with nearly self-similar scaling and vorticity growth of order 10710^7, and later exhibited (Foundations of Computational Mathematics, 2026) a nearly self-similar blowup for generalized axisymmetric Navier–Stokes equations in an effective fractional dimension ≈3.19\approx 3.19 — but no rigorous blowup or regularity result exists for the standard three-dimensional system.

Problem Statement

Consider the three-dimensional incompressible Navier–Stokes equations ∂tv+(v⋅∇)v−Δv+∇p=0,∇⋅v=0in R3×(0,∞)\partial_t v+(v\cdot\nabla)v-\Delta v+\nabla p=0,\qquad \nabla\cdot v=0\quad\text{in }\mathbb{R}^3\times(0,\infty) with viscosity normalized to 11. Let v0v_0 be smooth, divergence-free, of finite energy (i.e. v0∈L2(R3)v_0\in L^2(\mathbb{R}^3)), and axisymmetric, i.e. v0=v0r(r,z)er+v0θ(r,z)eθ+v03(r,z)e3v_0=v_0^r(r,z)e_r+v_0^{\theta}(r,z)e_\theta+v_0^3(r,z)e_3 in cylindrical coordinates (r,θ,z)(r,\theta,z), with nonzero swirl (v0θ≢0v_0^{\theta}\not\equiv 0) and rv0θ∈L∞r v_0^{\theta}\in L^\infty. The associated local-in-time strong (smooth) solution exists uniquely on some interval [0,Tmax⁡)[0,T_{\max}). Determine, without imposing any additional smallness, decay, form-boundedness, or structural condition on the swirl quantity Γ=rvθ\Gamma=r v^{\theta} beyond the class-defining hypotheses stated above (axisymmetry, nonzero swirl, and rv0θ∈L∞r v_0^{\theta}\in L^\infty):

  1. (Global regularity) Prove that Tmax⁡=∞T_{\max}=\infty for every such initial datum, i.e. the solution remains smooth with finite energy for all time; or

  2. (Finite-time blowup) Exhibit such an initial datum for which the solution loses smoothness at some finite time T<∞T<\infty, with a rigorous characterization of the blowup (e.g. ∥v(⋅,t)∥L∞→∞\|v(\cdot,t)\|_{L^\infty}\to\infty as t↑Tt\uparrow T).

Exactly one of the two alternatives must hold, and both are currently unproven. This is the classical axisymmetric-with-swirl open problem within the Clay Navier–Stokes millennium problem: alternative 2 would constitute a negative answer to the millennium problem for R3\mathbb{R}^3, and alternative 1 would be the first large-data global regularity result for a class retaining genuine three-dimensional vortex stretching.

References

  1. Zhen Lei and Qi S. Zhang, Criticality of the axially symmetric Navier–Stokes equations, Pacific Journal of Mathematics 289 (2017), no. 1, 169–187. DOI: 10.2140/pjm.2017.289.169; arXiv:1505.02628. https://arxiv.org/abs/1505.02628
  2. Qi S. Zhang and Xinghong Pan, A Review of Results on Axially Symmetric Navier-Stokes Equations, with Addendum by X. Pan and Q. Zhang, Analysis in Theory and Applications 38 (2022), 243–296. DOI: 10.4208/ata.oa-2020-0050; arXiv:2101.04905. https://arxiv.org/abs/2101.04905
  3. M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Journal of Applied Mathematics and Mechanics 32 (1968), no. 1, 52–62. DOI: 10.1016/0021-8928(68)90147-0. https://doi.org/10.1016/0021-8928(68)90147-0
  4. Chiun-Chuan Chen, Robert M. Strain, Horng-Tzer Yau and Tai-Peng Tsai, Lower Bound on the Blow-up Rate of the Axisymmetric Navier–Stokes Equations, International Mathematics Research Notices, Vol. 2008, Art. ID rnn016. DOI: 10.1093/imrn/rnn016. https://doi.org/10.1093/imrn/rnn016
  5. Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin and Vladimir Šverák, Liouville theorems for the Navier–Stokes equations and applications, Acta Mathematica 203 (2009), no. 1, 83–105. DOI: 10.1007/s11511-009-0039-6. https://doi.org/10.1007/s11511-009-0039-6
  6. Thomas Y. Hou, Potentially Singular Behavior of the 3D Navier–Stokes Equations, Foundations of Computational Mathematics 23 (2023), no. 6, 2251–2299. DOI: 10.1007/s10208-022-09578-4; arXiv:2107.06509. https://arxiv.org/abs/2107.06509
  7. Thomas Y. Hou, Nearly Self-similar Blowup of Generalized Axisymmetric Navier–Stokes Equations, Foundations of Computational Mathematics (2026). DOI: 10.1007/s10208-026-09748-8; arXiv:2405.10916. https://arxiv.org/abs/2405.10916
  8. Clay Mathematics Institute, Navier–Stokes Equation, Millennium Prize Problem. https://www.claymath.org/millennium/navier-stokes-equation/