Ninety years of attacks on the regularity question, and where to go deeper
The state of play
Since the 1930s, mathematicians have attacked the problem from many angles, from energy estimates and geometry to probability and computer-assisted analysis. The full 3D existence-and-smoothness question remains completely, stubbornly open.
But here's what people miss: we've learned an enormous amount from ninety years of failed attacks, and the collective picture is far richer than a simple "unsolved" label suggests. Entire strategies eliminated. Sub-cases closed. We know, with substantial progress: some subcases are resolved, several conditional criteria are understood, and major barriers are much clearer. What follows is a map of that progress.
Key milestones
Five results that reshaped the field:- 1934, Leray: Proved that global-in-time weak solutions exist for any reasonable initial data. Something persists forever. But does it stay smooth? That's the question Leray couldn't answer, and after ninety years, neither can anyone else.- 1982, Caffarelli, Kohn, Nirenberg: The set of possible singularities is extremely small: in the parabolic geometry natural to these equations, it has zero one-dimensional size. Vanishingly small. If blowup happens, it's sparse beyond imagination.- 1984, Beale, Kato, Majda: Huge result. A smooth solution can only break down if the vorticity blows up, which gave the entire field one precise target: control the relevant vorticity norm strongly enough, and a smooth solution cannot break down at that time.- 2016, Tao: Constructed blowup for an averaged Navier-Stokes sharing the same energy and scaling properties as the real thing, which means a proof for the real equation has to use finer structure than energy estimates and scaling alone. A barrier. Not a solution.- 2022, Albritton, Brué, Colombo: Leray-Hopf weak solutions aren't unique when you allow an external force. Bad news: the weakest solution class isn't as tame as we'd hoped, and this forces a rethinking of what "solution" even means at this level.
Dive deeper
This page is a map, not the territory. For the details:
Subproblems
The tractable pieces: 2D regularity, axisymmetric flows, critical spaces, and other special cases where real progress has been made.
Approaches
The major strategies mathematicians are pursuing: energy methods, harmonic analysis, probabilistic techniques, convex integration, and computational approaches.
About this page
Last reviewed: March 2026. This page is a living directory. As new results appear and deeper articles go up, it'll be updated.
Originally published on navier-stokes.org. The site covers the Navier-Stokes existence and smoothness problem with toggleable simple/rigorous explanations.
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