For the three-dimensional incompressible Navier–Stokes equations on the periodic torus T3T^3 (or on the whole space R3\mathbb{R}^3), Leray–Hopf weak solutions exist globally in time for every divergence-free initial velocity field with finite kinetic energy, but it is unknown whether such solutions always remain smooth. This is the existence-and-smoothness Clay Millennium Problem, originating in Leray's 1934 work; the source paper of this candidate (Cortissoz, Proc. Amer. Math. Soc. 2009) recalls it as the classical outstanding open problem left untouched by its small-data and borderline-space regularity criteria. The problem asks for either a proof that no finite-energy initial datum generates finite-time loss of regularity in any Leray–Hopf weak solution, or an explicit finite-energy datum (on R3\mathbb{R}^3 one may take smooth, rapidly decaying data) whose Leray–Hopf solution develops a singularity in finite time.

Background

The incompressible Navier–Stokes equations govern the velocity u(t,x)∈R3u(t,x)\in\mathbb{R}^3 and pressure p(t,x)p(t,x) of a viscous, constant-density fluid:

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

with viscosity ν>0\nu>0 (normalized to 11 by rescaling), posed either on the torus T3=[0,1]3T^3=[0,1]^3 with periodic boundary conditions or on the whole space R3\mathbb{R}^3. Leray (1934) proved that for every divergence-free initial velocity u0∈L2u_0\in L^2 (finite kinetic energy, 12∥u(t)∥L22<∞\tfrac{1}{2}\|u(t)\|_{L^2}^2<\infty) there exists a global-in-time weak solution in the class now called a Leray–Hopf weak solution: a distributional solution of the equations that has finite kinetic energy for all time, belongs to L∞(0,∞;L2)∩L2(0,∞;H1)L^\infty(0,\infty;L^2)\cap L^2(0,\infty;H^1) (where H1H^1 is the Sobolev space of L2L^2 functions with square-integrable gradient, representing finite enstrophy/dissipation), satisfies the energy inequality 12∥u(t)∥L22+∫0t∥∇u(s)∥L22ds≤12∥u0∥L22\tfrac{1}{2}\|u(t)\|_{L^2}^2+\int_0^t\|\nabla u(s)\|_{L^2}^2ds\le\tfrac{1}{2}\|u_0\|_{L^2}^2, and attains u0u_0 as initial state. The open question is whether these solutions are in fact always classical: smooth (C∞C^\infty jointly in space and time) and unique for t>0t>0, so that the initial-value problem is globally well-posed.

The central obstruction is supercriticality. The equations are invariant under the scaling u(t,x)↦uλ(t,x)=λu(λ2t,λx)u(t,x)\mapsto u_\lambda(t,x)=\lambda u(\lambda^2 t,\lambda x), and a norm is called critical if it is unchanged by this map; critical quantities (such as the spatial L3L^3 norm) are exactly those that decide regularity. The energy inequality controls only the L2L^2 norm, which becomes progressively weaker than any critical quantity as one zooms toward small spatial scales — precisely the scales on which a singularity would form. Consequently, all unconditional results are conditional or small-data: the Prodi–Serrin criteria state that a weak solution is smooth on (0,T)(0,T) if u∈Ls(0,T;Lr)u\in L^s(0,T;L^r) with 2s+3r=1\tfrac{2}{s}+\tfrac{3}{r}=1 and r>3r>3; the hardest endpoint r=3r=3 was settled by Escauriaza, Seregin and Šverák (2003), who proved that u∈L∞(0,T;L3)u\in L^\infty(0,T;L^3) already implies smoothness. Whether the required critical bounds hold for arbitrary data is unknown.

The source paper of this candidate, Cortissoz's "Some elementary estimates for the Navier-Stokes system" (Proc. Amer. Math. Soc. 137 (2009) 3343–3353), is a contribution to exactly this conditional program: on T3T^3 it proves that a weak solution that stays small in the borderline Fourier-side pseudo-measure space Φ(2)\Phi(2) (a space of sequences of Fourier coefficients with a weighted supremum norm, invariant under the equation's scaling at the L2L^2 energy level) is smooth, and it gives elementary proofs of some classical regularity criteria involving the Sobolev space H1/2(T3)H^{1/2}(T^3); the author's arXiv:0909.1699 addendum completes the iteration argument for the associated Fourier-transformed system. These are small-data and conditional criteria; the paper explicitly positions them against, and leaves open, the classical problem for data of arbitrary size.

Later work has sharpened the landscape without deciding the question. Tao (2016) proved finite-time blowup for a nonlinearly averaged (mollified) three-dimensional Navier–Stokes equation, showing that essentially all perturbative/iteration approaches face a genuine supercritical barrier — but only for a modified model equation. Buckmaster and Vicol (2019) used convex integration (a technique that constructs solutions by iteratively adding highly oscillatory, energy-correcting perturbations) to prove nonuniqueness of distributional weak solutions; their solutions live in a class strictly weaker than Leray–Hopf (they do not satisfy the Leray energy inequality), so nonuniqueness in the Leray–Hopf class, and regularity itself, remain open. The problem is one of the seven Clay Millennium Prize Problems, posed in 2000 with an official statement covering both R3\mathbb{R}^3 and the periodic setting, and the Clay Mathematics Institute currently lists it as unsolved.

Problem Statement

For the three-dimensional incompressible Navier–Stokes equations

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

posed on the periodic torus T3=[0,1]3T^3=[0,1]^3 and, in parallel, on the whole space R3\mathbb{R}^3, with initial velocity u(0)=u0u(0)=u_0 divergence-free and of arbitrary size in L2L^2 (finite kinetic energy; on R3\mathbb{R}^3, of particular interest is smooth, rapidly decaying u0u_0): determine whether every Leray–Hopf weak solution (distributional solution in L∞(0,∞;L2)∩L2(0,∞;H1)L^\infty(0,\infty;L^2)\cap L^2(0,\infty;H^1) satisfying the energy inequality and attaining u0u_0) is globally regular — smooth in space and time on (0,∞)×Ω(0,\infty)\times\Omega (where Ω\Omega is T3T^3 or R3\mathbb{R}^3 as posed above), and consequently unique in the Leray–Hopf class. The question is whether every such solution is globally regular, or whether there exists an explicit divergence-free u0∈L2u_0\in L^2 together with a rigorous construction of a Leray–Hopf solution that satisfies the energy inequality yet loses regularity at some finite time T∗T_*. This is the Clay Millennium existence-and-smoothness problem in its Leray formulation, stated as the outstanding open problem by the source paper, which leaves it open for data of arbitrary size. The official Clay formulation covers both R3\mathbb{R}^3 and the periodic torus, and the source paper itself recalls the problem on T3T^3.

References

  1. Jean Cortissoz, "Some elementary estimates for the Navier-Stokes system", Proc. Amer. Math. Soc. 137(10), 3343–3353 (2009). DOI: 10.1090/S0002-9939-09-09989-4. https://doi.org/10.1090/S0002-9939-09-09989-4
  2. Jean Cortissoz, "An addendum to the paper: 'Some elementary estimates for the Navier-Stokes system'", preprint (2009). arXiv:0909.1699. https://arxiv.org/abs/0909.1699
  3. Clay Mathematics Institute, "Navier–Stokes Equation", Millennium Problem official page (problem description by C. Fefferman), status "Unsolved", accessed 2026-08-27. https://www.claymath.org/millennium-problems/navier-stokes-equation
  4. J. Leray, "Sur le mouvement d'un liquide visqueux emplissant l'espace", Acta Mathematica 63, 193–248 (1934). DOI: 10.1007/BF02547354. https://doi.org/10.1007/BF02547354
  5. G. Prodi, "Un teorema di unicità per le equazioni di Navier-Stokes", Annali di Matematica Pura ed Applicata (4) 48, 173–182 (1959). DOI: 10.1007/BF02410664. https://doi.org/10.1007/BF02410664
  6. James Serrin, "On the interior regularity of weak solutions of the Navier-Stokes equations", Archive for Rational Mechanics and Analysis 9, 187–195 (1962). DOI: 10.1007/BF00253344. https://doi.org/10.1007/BF00253344
  7. L. Escauriaza, G. A. Seregin, V. Šverák, "L3,∞L_{3,\infty}-solutions of the Navier-Stokes equations and backward uniqueness", Russian Mathematical Surveys 58(2), 211–250 (2003). DOI: 10.1070/RM2003v058n02ABEH000609. https://doi.org/10.1070/RM2003v058n02ABEH000609
  8. Terence Tao, "Finite time blowup for an averaged three-dimensional Navier-Stokes equation", Journal of the American Mathematical Society 29(3), 601–674 (2016). DOI: 10.1090/jams/838. arXiv:1402.0290. https://arxiv.org/abs/1402.0290
  9. Tristan Buckmaster, Vlad Vicol, "Nonuniqueness of weak solutions to the Navier-Stokes equation", Annals of Mathematics (2) 189(1), 101–144 (2019). DOI: 10.4007/annals.2019.189.1.3. arXiv:1709.10033. https://arxiv.org/abs/1709.10033
  10. Myong-Hwan Ri, "Global regularity of Leray-Hopf weak solutions to 3D Navier-Stokes equations", preprint (2025). arXiv:2508.19590. https://arxiv.org/abs/2508.19590
  11. James Glimm, Jarret Petrillo, "Non-Smooth Solutions of the Navier-Stokes Equation and their Means", preprint (2024). arXiv:2410.09261. https://arxiv.org/abs/2410.09261
  12. James Glimm, Jarret Petrillo, "Smooth Solutions of the Navier-Stokes Equation", preprint (2025). arXiv:2505.13816. https://arxiv.org/abs/2505.13816