Progress on the Navier-Stokes Problem

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.

The Navier-Stokes regularity problem asks for global existence and smoothness of solutions to the 3D incompressible system with smooth, rapidly decaying data. It's resisted resolution since its modern formulation. The principal lines of attack include Leray–Hopf weak solution theory, partial regularity via geometric measure theory, conditional regularity through continuation criteria, probabilistic and stochastic methods, and convex integration for non-uniqueness.

No approach has closed the full problem. Together, though, they've clarified the critical barriers: supercritical scaling, the gap between energy-class and smooth solutions, and the disturbing possibility that uniqueness itself may fail in weak solution classes.

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.

• 1934, Leray: Global existence of weak solutions in \(L^2\) for divergence-free data \(u_0 \in L^2(\mathbb{R}^3)\), satisfying the energy inequality (J Math Pures Appl).
• 1982, Caffarelli–Kohn–Nirenberg (CKN): Partial regularity; \(\mathcal{P}^1(\mathrm{sing}\, u)=0\), meaning the one-dimensional parabolic Hausdorff measure of the singular set vanishes. Still the strongest general result we have (Comm Pure Appl Math).
• 1984, Beale–Kato–Majda (BKM): The continuation criterion that reshaped the field: a smooth solution on \([0,T)\) extends past \(T\) if and only if \(\int_0^T \|\omega(\cdot,t)\|_{L^\infty}\,dt < \infty\), reducing regularity to vorticity control. Originally for Euler; adapted to Navier-Stokes (Comm Math Phys).
• 2016, Tao: Finite-time blowup for an averaged Navier-Stokes equation obeying the same energy and scaling properties as the true system, meaning any regularity proof must exploit finer structure (J Amer Math Soc).
• 2022, Albritton–Brué–Colombo: Non-uniqueness of Leray-Hopf solutions for forced 3D Navier-Stokes, constructed via unstable self-similar solutions (Ann of Math).

This page is a map, not the territory. For the details:

For detailed treatment of specific research directions:

• Subproblems: resolved and partially resolved cases, including 2D global regularity (Ladyzhenskaya 1959), axisymmetric without swirl, critical-space results (\(L^3\), \(\dot{H}^{1/2}\), \(BMO^{-1}\)), and conditional regularity criteria beyond BKM.
• Approaches: the main proof strategies under active investigation, including energy and enstrophy methods, profile decomposition, mild solution theory, stochastic Navier-Stokes, convex integration programmes, and computational-assisted bounds.

Last reviewed: March 2026. This page is a living directory. As new results appear and deeper articles go up, it'll be updated.

Last reviewed: March 2026. This page serves as a navigational hub. For corrections or suggestions, see the site repository.