Global regularity of the three-dimensional Navier–Stokes equations
Let $\mathbb{T}^3 = (\mathbb{R}/\mathbb{Z})^3$ and $\nu > 0$, and let $u_0 \in C^\infty(\mathbb{T}^3; \mathbb{R}^3)$ be divergence-free with zero mean. Consider the unforced initial-value problem $$\partial_t u + (u \cdot \nabla) u + \nabla p = \nu \Delta u, \qquad \nabla \cdot u = 0, \qquad u(0) = u_0,$$ with the pressure normalized to have zero spatial mean. A smooth solution exists on a time interval depending on $u_0$, and for all time when $u_0$ is small (see Fefferman). Leray constructed global weak solutions in $\mathbb{R}^3$ (1934), and Caffarelli, Kohn and Nirenberg proved that the singular set of a suitable weak solution has zero one-dimensional parabolic Hausdorff measure (1982). Tao proved finite-time blowup for an averaged Navier–Stokes equation that obeys the same energy identity (2016). The question below is one of four alternative statements in Fefferman's description of the Clay Millennium Prize problem; the two breakdown statements there allow an external force.
Problem. Does the problem have a smooth solution on $\mathbb{T}^3 \times [0, T]$ for every finite $T > 0$ and every such $u_0$? A negative answer must exhibit an admissible $u_0$ for which smooth continuation fails, at fixed positive viscosity and with no external force.
Source: Leray (1934); in this form, the periodic existence-and-smoothness statement (B) of Fefferman's description of the Clay Millennium Prize problem (2000; published 2006). Stated by AIM — Open Applied Problems with zero-mean data and pressure.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, global regularity, incompressible flows, Navier–Stokes equations
References
@article{leray1934,
author = {Leray, Jean},
title = {Sur le mouvement d'un liquide visqueux emplissant l'espace},
journal = {Acta Math.},
volume = {63},
number = {1},
year = {1934},
pages = {193--248},
doi = {10.1007/BF02547354},
}
@incollection{fefferman2006,
author = {Fefferman, Charles L.},
title = {Existence and smoothness of the {N}avier--{S}tokes equation},
booktitle = {The Millennium Prize Problems},
editor = {Carlson, James and Jaffe, Arthur and Wiles, Andrew},
publisher = {Clay Mathematics Institute and American Mathematical Society},
address = {Cambridge, MA and Providence, RI},
year = {2006},
pages = {57--67},
}
@article{caffarelli1982,
author = {Caffarelli, L. and Kohn, R. and Nirenberg, L.},
title = {Partial regularity of suitable weak solutions of the {N}avier--{S}tokes equations},
journal = {Comm. Pure Appl. Math.},
volume = {35},
number = {6},
year = {1982},
pages = {771--831},
doi = {10.1002/cpa.3160350604},
}
@article{tao2016,
author = {Tao, Terence},
title = {Finite time blowup for an averaged three-dimensional {N}avier--{S}tokes equation},
journal = {J. Amer. Math. Soc.},
volume = {29},
number = {3},
year = {2016},
pages = {601--674},
doi = {10.1090/jams/838},
eprint = {1402.0290},
archivePrefix = {arXiv},
}
@misc{aim-navier-stokes-global-regularity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global regularity of the three-dimensional Navier–Stokes equations}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/071-navier-stokes-global-regularity.md},
note = {Entry 071 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00618, "Global regularity of the three-dimensional Navier–Stokes equations", clopen.solutions.