Weak solutions, regularity criteria, and the main proof strategies
Energy methods and Leray-Hopf theory
The oldest approach starts with energy. A moving fluid carries kinetic energy, and viscosity eats it, like friction grinding things to a halt. Total energy can only decrease over time, assuming nothing's pumping energy in from outside.
Leray saw this in 1934 and made a key move: use the energy bound to prove that a global weak solution with finite kinetic energy has to exist. Build approximate solutions, artificially smoothed. Show they all obey the energy bound. Take a limit. Something must survive in that limit, and it does.
But here's the catch. Energy bounds are blunt instruments. They guarantee the fluid has finite total energy, sure, but they can't tell you the velocity stays finite at every single point in space and time. That gap between "finite energy" and "smooth everywhere" is exactly the regularity problem, and it's been open for ninety years.
Paper links: Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace (1934); Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen (1951).
CKN partial regularity
The Caffarelli-Kohn-Nirenberg approach (1982) doesn't try to prove full smoothness. It asks something else entirely: how bad can the singularities actually be?
Barely bad at all. Their $\varepsilon$-regularity theorem says that if certain scale-invariant local quantities are small enough in a small space-time region, the solution is automatically smooth there. And since total energy is finite, there simply isn't enough "budget" for many singular points to coexist.
Think of it this way. A wall might have cracks. But the total length of all those cracks combined is zero, meaning the singular set is extremely small in the parabolic measure-theoretic sense (one-dimensional parabolic Hausdorff measure zero).
Paper links: Caffarelli-Kohn-Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations (1982); Albritton-Barker-Prange, Epsilon regularity for the Navier-Stokes equations via weak-strong uniqueness.
Beale-Kato-Majda and vorticity control
Here's a sharp reduction of the whole problem. Beale, Kato, and Majda proved in 1984 that for the 3D Euler equations, blowup can only happen if vorticity control is lost. Analogous criteria were later established for Navier-Stokes. That's it. One condition.
Vorticity measures local spin. The BKM criterion says: keep the maximum spin bounded in the right norm, and the solution stays smooth. Everything else falls in line automatically.
One family of quantities to control. Unfortunately, actually controlling them has turned out to be exactly as hard as the original problem. The reduction is clean. The execution remains out of reach.
Paper links: Beale-Kato-Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations (1984); Kozono-Taniuchi, Bilinear estimates in BMO and the Navier-Stokes equations (2000).
Critical and subcritical spaces
A more modern angle works with function spaces (like $L^3$ or $\dot{H}^{1/2}$) that sit right at the boundary of what the scaling symmetry allows. These are critical spaces, and they are where the sharpest regularity results live.
The logic is clean: if you can show a solution stays within certain critical-space bounds, smoothness follows automatically. Multiple teams have proved this, building a whole menu of regularity criteria (conditions that guarantee smoothness if you can verify them).
The problem is the gap. We can prove subcritical bounds from energy methods. We need critical bounds. That gap is narrow, sometimes a single derivative of regularity, but it has resisted every attempt to close it.
Paper links: Koch-Tataru, Well-posedness for the Navier-Stokes equations (2001); Kenig-Koch, An alternative approach to regularity for the Navier-Stokes equations in critical spaces; Gallagher-Koch-Planchon, A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion. For a detailed comparison of why energy criticality succeeds in 2D but fails in 3D, see Why 2D Is Easier Than 3D.
Harmonic analysis and Littlewood-Paley
Modern PDE theory borrows heavily from harmonic analysis. The core idea: break a function into waves at different frequencies, the way you'd split a musical chord into individual notes. Except here, the "notes" are spatial oscillations of fluid velocity at wildly different scales.
Littlewood-Paley decomposition does exactly this. Chop the velocity field into scale-by-scale components. Track how energy flows between them. Suddenly the informal physical intuition of "energy cascade" becomes something you can actually prove theorems about, and the theorems are precise. These methods have produced many of the sharpest results on regularity criteria and blowup rates.
Paper links: Cannone-Meyer, Littlewood-Paley decomposition and Navier-Stokes equations (1995); Gallagher-Koch-Planchon, A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier-Stokes regularity criterion.
Geometric and topological methods
Here's a different instinct entirely. Instead of tracking numbers (norms, energies), these methods study the shape of the solution: how vortex tubes bend, how regions of intense rotation arrange themselves in space.
The key insight is that blowup isn't just about something getting big. It's about the fluid organizing itself into a very specific geometric configuration. If you can show that configuration is impossible (because it contradicts the energy-dissipation structure, or incompressibility, or both), you've ruled out blowup without ever computing a norm.
This geometric viewpoint has grown into a viewpoint that has inspired several rigorous regularity criteria alongside purely analytic methods. And it feels different. It asks what shape does disaster take? instead of how big can this number get?
Paper links: Constantin-Fefferman, Geometric constraints on potentially singular solutions for the 3-D Euler equations (1993); Albritton-Barker-Prange, Localized smoothing and concentration for the Navier-Stokes equations in the half space.
Non-uniqueness and convex integration
This one caught people off guard. The weak solutions from Leray's method (Section 1) turn out to be non-unique, at least when external forcing is present.
The weapon is convex integration, a technique originally built for geometry problems and adapted to fluid equations by De Lellis and Székelyhidi starting around 2009. The idea: construct "wild" solutions by iteratively piling on high-frequency corrections that collectively satisfy the equation but behave erratically.
For 3D Euler (Navier-Stokes without viscosity), Buckmaster and Vicol (2019) proved weak solutions aren't unique. Then in 2022, Albritton, Brué, and Colombo proved that even Leray-Hopf solutions of 3D Navier-Stokes are non-unique when external force is present. Whether non-uniqueness persists for the unforced Navier-Stokes equations remains open.
Why does this matter? Because "a weak solution exists" has been the headline result since 1934. Now we know it doesn't pin down a single answer. The question sharpens: which solution, if any, is the physically correct one?
Paper links: De Lellis-Székelyhidi, Dissipative continuous Euler flows (2013); Buckmaster-Vicol, Nonuniqueness of weak solutions to the Navier-Stokes equation (2019); Albritton-Brué-Colombo, Non-uniqueness of Leray solutions of the forced Navier-Stokes equations (2022).
Proof barriers and supercritical blowup
Can we at least rule out certain proof strategies? Terence Tao showed in 2016 that yes, we can. And the result is sobering.
Tao built a modified version of the Navier-Stokes equations, an "averaged" system, that keeps many key structural features of the real equations: the energy identity, the way enstrophy (a measure of vorticity intensity) grows, the scaling symmetry. But in this modified system, solutions blow up in finite time.
The implication rules out broad families of proof strategies. Any proof that global smoothness holds for the real equations must use some specific structural property of the true nonlinearity that the averaged system doesn't have. You can't prove regularity using only energy bounds, scaling, and enstrophy growth. Those tools alone are consistent with blowup.
This doesn't say the real equations blow up. It says entire families of proof strategies are dead ends. The eventual proof (if regularity holds) must be sharper than a generic energy argument. Much sharper.
Paper links: Tao, Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2016).
Continue exploring
This article is part of Progress.
From Leray's 1934 existence proof through convex integration and Tao's proof barriers, these are the main strategies people have thrown at the 3D Navier-Stokes problem. None has resolved the full 3D regularity problem. For context on how viscosity shapes the mathematics compared to the inviscid Euler equations, see Euler vs. Navier-Stokes. For the current status, see Is the Navier-Stokes Problem Solved? For the exact formal statement, return to The Millennium Problem.
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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